3.16.65 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\) [1565]

Optimal. Leaf size=446 \[ \frac {3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{65536 c^6}-\frac {\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac {3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{131072 c^{13/2}} \]

[Out]

-1/8192*(-4*a*c+b^2)^2*e*(40*c^2*d^2+11*b^2*e^2-4*c*e*(a*e+10*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5+1/2560*(
-4*a*c+b^2)*e*(40*c^2*d^2+11*b^2*e^2-4*c*e*(a*e+10*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^4+1/30*(-b*e+2*c*d)*(
e*x+d)^2*(c*x^2+b*x+a)^(7/2)/c+1/5*(e*x+d)^3*(c*x^2+b*x+a)^(7/2)+1/6720*(128*c^3*d^3-99*b^3*e^3+4*b*c*e^2*(97*
a*e+90*b*d)-8*c^2*d*e*(160*a*e+17*b*d)+14*c*e*(8*c^2*d^2+11*b^2*e^2-4*c*e*(9*a*e+2*b*d))*x)*(c*x^2+b*x+a)^(7/2
)/c^3-3/131072*(-4*a*c+b^2)^4*e*(40*c^2*d^2+11*b^2*e^2-4*c*e*(a*e+10*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^
2+b*x+a)^(1/2))/c^(13/2)+3/65536*(-4*a*c+b^2)^3*e*(40*c^2*d^2+11*b^2*e^2-4*c*e*(a*e+10*b*d))*(2*c*x+b)*(c*x^2+
b*x+a)^(1/2)/c^6

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Rubi [A]
time = 0.40, antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {846, 793, 626, 635, 212} \begin {gather*} -\frac {3 e \left (b^2-4 a c\right )^4 \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{131072 c^{13/2}}+\frac {3 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{65536 c^6}-\frac {e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{8192 c^5}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{2560 c^4}+\frac {\left (a+b x+c x^2\right )^{7/2} \left (14 c e x \left (-4 c e (9 a e+2 b d)+11 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (160 a e+17 b d)+4 b c e^2 (97 a e+90 b d)-99 b^3 e^3+128 c^3 d^3\right )}{6720 c^3}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{30 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(3*(b^2 - 4*a*c)^3*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(6553
6*c^6) - ((b^2 - 4*a*c)^2*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/
2))/(8192*c^5) + ((b^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^
2)^(5/2))/(2560*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(30*c) + ((d + e*x)^3*(a + b*x + c*
x^2)^(7/2))/5 + ((128*c^3*d^3 - 99*b^3*e^3 + 4*b*c*e^2*(90*b*d + 97*a*e) - 8*c^2*d*e*(17*b*d + 160*a*e) + 14*c
*e*(8*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(2*b*d + 9*a*e))*x)*(a + b*x + c*x^2)^(7/2))/(6720*c^3) - (3*(b^2 - 4*a*c)^
4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
131072*c^(13/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\int (d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\int (d+e x) \left (\frac {3}{2} c \left (7 b^2 d e-44 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac {3}{2} c \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}+\frac {\left (3 \left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{640 c^3}\\ &=\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac {\left (\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{16384 c^5}\\ &=\frac {3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{65536 c^6}-\frac {\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac {\left (3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{131072 c^6}\\ &=\frac {3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{65536 c^6}-\frac {\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac {\left (3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{65536 c^6}\\ &=\frac {3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{65536 c^6}-\frac {\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac {(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac {1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac {3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{131072 c^{13/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(927\) vs. \(2(446)=892\).
time = 7.14, size = 927, normalized size = 2.08 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (3465 b^9 e^3-210 b^8 c e^2 (60 d+11 e x)-640 b^4 c^5 e x^3 \left (9 d^2+8 d e x+2 e^2 x^2\right )+168 b^7 c^2 e \left (75 d^2+50 d e x+11 e^2 x^2\right )+64 b^5 c^4 e x^2 \left (105 d^2+90 d e x+22 e^2 x^2\right )-48 b^6 c^3 e x \left (175 d^2+140 d e x+33 e^2 x^2\right )+16384 c^9 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )+5120 b^3 c^6 x^3 \left (384 d^3+897 d^2 e x+734 d e^2 x^2+207 e^3 x^3\right )+8192 b c^8 x^5 \left (720 d^3+1845 d^2 e x+1610 d e^2 x^2+476 e^3 x^3\right )+2048 b^2 c^7 x^4 \left (2880 d^3+7125 d^2 e x+6060 d e^2 x^2+1757 e^3 x^3\right )-1280 a^4 c^4 e^2 (-449 b e+2 c (512 d+63 e x))+1280 a^3 c^3 \left (-537 b^3 e^3+62 b^2 c e^2 (27 d+4 e x)-2 b c^2 e \left (837 d^2+374 d e x+65 e^2 x^2\right )+4 c^3 \left (384 d^3+315 d^2 e x+128 d e^2 x^2+21 e^3 x^3\right )\right )+96 a^2 c^2 \left (3003 b^5 e^3-10 b^4 c e^2 (1022 d+167 e x)-40 b^2 c^3 e x \left (141 d^2+92 d e x+19 e^2 x^2\right )+20 b^3 c^2 e \left (511 d^2+282 d e x+55 e^2 x^2\right )+160 b c^4 x \left (384 d^3+663 d^2 e x+454 d e^2 x^2+114 e^3 x^3\right )+64 c^5 x^2 \left (960 d^3+2065 d^2 e x+1600 d e^2 x^2+434 e^3 x^3\right )\right )+16 a c \left (-3255 b^7 e^3+42 b^6 c e^2 (275 d+48 e x)-160 b^3 c^4 e x^2 \left (33 d^2+26 d e x+6 e^2 x^2\right )+20 b^4 c^3 e x \left (357 d^2+264 d e x+59 e^2 x^2\right )-6 b^5 c^2 e \left (1925 d^2+1190 d e x+249 e^2 x^2\right )+960 b^2 c^5 x^2 \left (384 d^3+815 d^2 e x+628 d e^2 x^2+170 e^3 x^3\right )+512 c^7 x^4 \left (720 d^3+1785 d^2 e x+1520 d e^2 x^2+441 e^3 x^3\right )+256 b c^6 x^3 \left (2880 d^3+6765 d^2 e x+5550 d e^2 x^2+1567 e^3 x^3\right )\right )\right )+315 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{13762560 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3465*b^9*e^3 - 210*b^8*c*e^2*(60*d + 11*e*x) - 640*b^4*c^5*e*x^3*(9*d^2 + 8*
d*e*x + 2*e^2*x^2) + 168*b^7*c^2*e*(75*d^2 + 50*d*e*x + 11*e^2*x^2) + 64*b^5*c^4*e*x^2*(105*d^2 + 90*d*e*x + 2
2*e^2*x^2) - 48*b^6*c^3*e*x*(175*d^2 + 140*d*e*x + 33*e^2*x^2) + 16384*c^9*x^6*(120*d^3 + 315*d^2*e*x + 280*d*
e^2*x^2 + 84*e^3*x^3) + 5120*b^3*c^6*x^3*(384*d^3 + 897*d^2*e*x + 734*d*e^2*x^2 + 207*e^3*x^3) + 8192*b*c^8*x^
5*(720*d^3 + 1845*d^2*e*x + 1610*d*e^2*x^2 + 476*e^3*x^3) + 2048*b^2*c^7*x^4*(2880*d^3 + 7125*d^2*e*x + 6060*d
*e^2*x^2 + 1757*e^3*x^3) - 1280*a^4*c^4*e^2*(-449*b*e + 2*c*(512*d + 63*e*x)) + 1280*a^3*c^3*(-537*b^3*e^3 + 6
2*b^2*c*e^2*(27*d + 4*e*x) - 2*b*c^2*e*(837*d^2 + 374*d*e*x + 65*e^2*x^2) + 4*c^3*(384*d^3 + 315*d^2*e*x + 128
*d*e^2*x^2 + 21*e^3*x^3)) + 96*a^2*c^2*(3003*b^5*e^3 - 10*b^4*c*e^2*(1022*d + 167*e*x) - 40*b^2*c^3*e*x*(141*d
^2 + 92*d*e*x + 19*e^2*x^2) + 20*b^3*c^2*e*(511*d^2 + 282*d*e*x + 55*e^2*x^2) + 160*b*c^4*x*(384*d^3 + 663*d^2
*e*x + 454*d*e^2*x^2 + 114*e^3*x^3) + 64*c^5*x^2*(960*d^3 + 2065*d^2*e*x + 1600*d*e^2*x^2 + 434*e^3*x^3)) + 16
*a*c*(-3255*b^7*e^3 + 42*b^6*c*e^2*(275*d + 48*e*x) - 160*b^3*c^4*e*x^2*(33*d^2 + 26*d*e*x + 6*e^2*x^2) + 20*b
^4*c^3*e*x*(357*d^2 + 264*d*e*x + 59*e^2*x^2) - 6*b^5*c^2*e*(1925*d^2 + 1190*d*e*x + 249*e^2*x^2) + 960*b^2*c^
5*x^2*(384*d^3 + 815*d^2*e*x + 628*d*e^2*x^2 + 170*e^3*x^3) + 512*c^7*x^4*(720*d^3 + 1785*d^2*e*x + 1520*d*e^2
*x^2 + 441*e^3*x^3) + 256*b*c^6*x^3*(2880*d^3 + 6765*d^2*e*x + 5550*d*e^2*x^2 + 1567*e^3*x^3))) + 315*(b^2 - 4
*a*c)^4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(
13762560*c^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2152\) vs. \(2(412)=824\).
time = 1.19, size = 2153, normalized size = 4.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(1812\)
default	\(\text {Expression too large to display}\)	\(2153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*c*e^3*(1/10*x^3*(c*x^2+b*x+a)^(7/2)/c-13/20*b/c*(1/9*x^2*(c*x^2+b*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a
)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c
*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/
c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*
a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4
*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-2/9*a/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(
1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/
c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
)-3/10*a/c*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x
+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b
*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*
(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b
)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))))+(b*e^3+6*c*d*
e^2)*(1/9*x^2*(c*x^2+b*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c
-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4
*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)
/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))))))-2/9*a/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*
a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4
*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))))+(3*b*d*e^2+6*c*d^2*e)*(1/8*x*(c*x^2+b*x+a)^
(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(
1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^
(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*
c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a
*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+(3*b*d^2*e+2*c*d^3)*(1/7*(c*x^2+b*x+a)^(7/2)/c
-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4
*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))))))+b*d^3*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c
+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2)))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1152 vs. \(2 (426) = 852\).
time = 5.34, size = 2307, normalized size = 5.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/27525120*(315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b^2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^
9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 256*a^3*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a
^2*b^6*c^2 - 3200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 -
4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(1966080*c^10*d^3*x^6 + 5898240*b*c^9*d^3*x^5 + 5898
240*a^2*b*c^7*d^3*x + 1966080*a^3*c^7*d^3 + 5898240*(b^2*c^8 + a*c^9)*d^3*x^4 + 1966080*(b^3*c^7 + 6*a*b*c^8)*
d^3*x^3 + 5898240*(a*b^2*c^7 + a^2*c^8)*d^3*x^2 + (1376256*c^10*x^9 + 3899392*b*c^9*x^8 + 3465*b^9*c - 52080*a
*b^7*c^2 + 288288*a^2*b^5*c^3 - 687360*a^3*b^3*c^4 + 574720*a^4*b*c^5 + 14336*(251*b^2*c^8 + 252*a*c^9)*x^7 +
1024*(1035*b^3*c^7 + 6268*a*b*c^8)*x^6 - 256*(5*b^4*c^6 - 10200*a*b^2*c^7 - 10416*a^2*c^8)*x^5 + 128*(11*b^5*c
^5 - 120*a*b^3*c^6 + 13680*a^2*b*c^7)*x^4 - 16*(99*b^6*c^4 - 1180*a*b^4*c^5 + 4560*a^2*b^2*c^6 - 6720*a^3*c^7)
*x^3 + 8*(231*b^7*c^3 - 2988*a*b^5*c^4 + 13200*a^2*b^3*c^5 - 20800*a^3*b*c^6)*x^2 - 2*(1155*b^8*c^2 - 16128*a*
b^6*c^3 + 80160*a^2*b^4*c^4 - 158720*a^3*b^2*c^5 + 80640*a^4*c^6)*x)*e^3 + 40*(114688*c^10*d*x^8 + 329728*b*c^
9*d*x^7 + 1024*(303*b^2*c^8 + 304*a*c^9)*d*x^6 + 256*(367*b^3*c^7 + 2220*a*b*c^8)*d*x^5 - 128*(b^4*c^6 - 1884*
a*b^2*c^7 - 1920*a^2*c^8)*d*x^4 + 16*(9*b^5*c^5 - 104*a*b^3*c^6 + 10896*a^2*b*c^7)*d*x^3 - 8*(21*b^6*c^4 - 264
*a*b^4*c^5 + 1104*a^2*b^2*c^6 - 2048*a^3*c^7)*d*x^2 + 2*(105*b^7*c^3 - 1428*a*b^5*c^4 + 6768*a^2*b^3*c^5 - 119
68*a^3*b*c^6)*d*x - (315*b^8*c^2 - 4620*a*b^6*c^3 + 24528*a^2*b^4*c^4 - 53568*a^3*b^2*c^5 + 32768*a^4*c^6)*d)*
e^2 + 120*(43008*c^10*d^2*x^7 + 125952*b*c^9*d^2*x^6 + 256*(475*b^2*c^8 + 476*a*c^9)*d^2*x^5 + 128*(299*b^3*c^
7 + 1804*a*b*c^8)*d^2*x^4 - 16*(3*b^4*c^6 - 6520*a*b^2*c^7 - 6608*a^2*c^8)*d^2*x^3 + 8*(7*b^5*c^5 - 88*a*b^3*c
^6 + 10608*a^2*b*c^7)*d^2*x^2 - 2*(35*b^6*c^4 - 476*a*b^4*c^5 + 2256*a^2*b^2*c^6 - 6720*a^3*c^7)*d^2*x + (105*
b^7*c^3 - 1540*a*b^5*c^4 + 8176*a^2*b^3*c^5 - 17856*a^3*b*c^6)*d^2)*e)*sqrt(c*x^2 + b*x + a))/c^7, 1/13762560*
(315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b^2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^9*c - 16*a*b^
7*c^2 + 96*a^2*b^5*c^3 - 256*a^3*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a^2*b^6*c^2 -
3200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)
*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(1966080*c^10*d^3*x^6 + 5898240*b*c^9*d^3*x^5 + 5898240*a^2*b*c^7*d^3*x
+ 1966080*a^3*c^7*d^3 + 5898240*(b^2*c^8 + a*c^9)*d^3*x^4 + 1966080*(b^3*c^7 + 6*a*b*c^8)*d^3*x^3 + 5898240*(
a*b^2*c^7 + a^2*c^8)*d^3*x^2 + (1376256*c^10*x^9 + 3899392*b*c^9*x^8 + 3465*b^9*c - 52080*a*b^7*c^2 + 288288*a
^2*b^5*c^3 - 687360*a^3*b^3*c^4 + 574720*a^4*b*c^5 + 14336*(251*b^2*c^8 + 252*a*c^9)*x^7 + 1024*(1035*b^3*c^7
+ 6268*a*b*c^8)*x^6 - 256*(5*b^4*c^6 - 10200*a*b^2*c^7 - 10416*a^2*c^8)*x^5 + 128*(11*b^5*c^5 - 120*a*b^3*c^6
+ 13680*a^2*b*c^7)*x^4 - 16*(99*b^6*c^4 - 1180*a*b^4*c^5 + 4560*a^2*b^2*c^6 - 6720*a^3*c^7)*x^3 + 8*(231*b^7*c
^3 - 2988*a*b^5*c^4 + 13200*a^2*b^3*c^5 - 20800*a^3*b*c^6)*x^2 - 2*(1155*b^8*c^2 - 16128*a*b^6*c^3 + 80160*a^2
*b^4*c^4 - 158720*a^3*b^2*c^5 + 80640*a^4*c^6)*x)*e^3 + 40*(114688*c^10*d*x^8 + 329728*b*c^9*d*x^7 + 1024*(303
*b^2*c^8 + 304*a*c^9)*d*x^6 + 256*(367*b^3*c^7 + 2220*a*b*c^8)*d*x^5 - 128*(b^4*c^6 - 1884*a*b^2*c^7 - 1920*a^
2*c^8)*d*x^4 + 16*(9*b^5*c^5 - 104*a*b^3*c^6 + 10896*a^2*b*c^7)*d*x^3 - 8*(21*b^6*c^4 - 264*a*b^4*c^5 + 1104*a
^2*b^2*c^6 - 2048*a^3*c^7)*d*x^2 + 2*(105*b^7*c^3 - 1428*a*b^5*c^4 + 6768*a^2*b^3*c^5 - 11968*a^3*b*c^6)*d*x -
(315*b^8*c^2 - 4620*a*b^6*c^3 + 24528*a^2*b^4*c^4 - 53568*a^3*b^2*c^5 + 32768*a^4*c^6)*d)*e^2 + 120*(43008*c^
10*d^2*x^7 + 125952*b*c^9*d^2*x^6 + 256*(475*b^2*c^8 + 476*a*c^9)*d^2*x^5 + 128*(299*b^3*c^7 + 1804*a*b*c^8)*d
^2*x^4 - 16*(3*b^4*c^6 - 6520*a*b^2*c^7 - 6608*a^2*c^8)*d^2*x^3 + 8*(7*b^5*c^5 - 88*a*b^3*c^6 + 10608*a^2*b*c^
7)*d^2*x^2 - 2*(35*b^6*c^4 - 476*a*b^4*c^5 + 2256*a^2*b^2*c^6 - 6720*a^3*c^7)*d^2*x + (105*b^7*c^3 - 1540*a*b^
5*c^4 + 8176*a^2*b^3*c^5 - 17856*a^3*b*c^6)*d^2)*e)*sqrt(c*x^2 + b*x + a))/c^7]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (426) = 852\).
time = 3.66, size = 1302, normalized size = 2.92 \begin {gather*} \frac {1}{6881280} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, {\left (6 \, c^{3} x e^{3} + \frac {20 \, c^{12} d e^{2} + 17 \, b c^{11} e^{3}}{c^{9}}\right )} x + \frac {360 \, c^{12} d^{2} e + 920 \, b c^{11} d e^{2} + 251 \, b^{2} c^{10} e^{3} + 252 \, a c^{11} e^{3}}{c^{9}}\right )} x + \frac {1920 \, c^{12} d^{3} + 14760 \, b c^{11} d^{2} e + 12120 \, b^{2} c^{10} d e^{2} + 12160 \, a c^{11} d e^{2} + 1035 \, b^{3} c^{9} e^{3} + 6268 \, a b c^{10} e^{3}}{c^{9}}\right )} x + \frac {23040 \, b c^{11} d^{3} + 57000 \, b^{2} c^{10} d^{2} e + 57120 \, a c^{11} d^{2} e + 14680 \, b^{3} c^{9} d e^{2} + 88800 \, a b c^{10} d e^{2} - 5 \, b^{4} c^{8} e^{3} + 10200 \, a b^{2} c^{9} e^{3} + 10416 \, a^{2} c^{10} e^{3}}{c^{9}}\right )} x + \frac {46080 \, b^{2} c^{10} d^{3} + 46080 \, a c^{11} d^{3} + 35880 \, b^{3} c^{9} d^{2} e + 216480 \, a b c^{10} d^{2} e - 40 \, b^{4} c^{8} d e^{2} + 75360 \, a b^{2} c^{9} d e^{2} + 76800 \, a^{2} c^{10} d e^{2} + 11 \, b^{5} c^{7} e^{3} - 120 \, a b^{3} c^{8} e^{3} + 13680 \, a^{2} b c^{9} e^{3}}{c^{9}}\right )} x + \frac {122880 \, b^{3} c^{9} d^{3} + 737280 \, a b c^{10} d^{3} - 360 \, b^{4} c^{8} d^{2} e + 782400 \, a b^{2} c^{9} d^{2} e + 792960 \, a^{2} c^{10} d^{2} e + 360 \, b^{5} c^{7} d e^{2} - 4160 \, a b^{3} c^{8} d e^{2} + 435840 \, a^{2} b c^{9} d e^{2} - 99 \, b^{6} c^{6} e^{3} + 1180 \, a b^{4} c^{7} e^{3} - 4560 \, a^{2} b^{2} c^{8} e^{3} + 6720 \, a^{3} c^{9} e^{3}}{c^{9}}\right )} x + \frac {737280 \, a b^{2} c^{9} d^{3} + 737280 \, a^{2} c^{10} d^{3} + 840 \, b^{5} c^{7} d^{2} e - 10560 \, a b^{3} c^{8} d^{2} e + 1272960 \, a^{2} b c^{9} d^{2} e - 840 \, b^{6} c^{6} d e^{2} + 10560 \, a b^{4} c^{7} d e^{2} - 44160 \, a^{2} b^{2} c^{8} d e^{2} + 81920 \, a^{3} c^{9} d e^{2} + 231 \, b^{7} c^{5} e^{3} - 2988 \, a b^{5} c^{6} e^{3} + 13200 \, a^{2} b^{3} c^{7} e^{3} - 20800 \, a^{3} b c^{8} e^{3}}{c^{9}}\right )} x + \frac {2949120 \, a^{2} b c^{9} d^{3} - 4200 \, b^{6} c^{6} d^{2} e + 57120 \, a b^{4} c^{7} d^{2} e - 270720 \, a^{2} b^{2} c^{8} d^{2} e + 806400 \, a^{3} c^{9} d^{2} e + 4200 \, b^{7} c^{5} d e^{2} - 57120 \, a b^{5} c^{6} d e^{2} + 270720 \, a^{2} b^{3} c^{7} d e^{2} - 478720 \, a^{3} b c^{8} d e^{2} - 1155 \, b^{8} c^{4} e^{3} + 16128 \, a b^{6} c^{5} e^{3} - 80160 \, a^{2} b^{4} c^{6} e^{3} + 158720 \, a^{3} b^{2} c^{7} e^{3} - 80640 \, a^{4} c^{8} e^{3}}{c^{9}}\right )} x + \frac {1966080 \, a^{3} c^{9} d^{3} + 12600 \, b^{7} c^{5} d^{2} e - 184800 \, a b^{5} c^{6} d^{2} e + 981120 \, a^{2} b^{3} c^{7} d^{2} e - 2142720 \, a^{3} b c^{8} d^{2} e - 12600 \, b^{8} c^{4} d e^{2} + 184800 \, a b^{6} c^{5} d e^{2} - 981120 \, a^{2} b^{4} c^{6} d e^{2} + 2142720 \, a^{3} b^{2} c^{7} d e^{2} - 1310720 \, a^{4} c^{8} d e^{2} + 3465 \, b^{9} c^{3} e^{3} - 52080 \, a b^{7} c^{4} e^{3} + 288288 \, a^{2} b^{5} c^{5} e^{3} - 687360 \, a^{3} b^{3} c^{6} e^{3} + 574720 \, a^{4} b c^{7} e^{3}}{c^{9}}\right )} + \frac {3 \, {\left (40 \, b^{8} c^{2} d^{2} e - 640 \, a b^{6} c^{3} d^{2} e + 3840 \, a^{2} b^{4} c^{4} d^{2} e - 10240 \, a^{3} b^{2} c^{5} d^{2} e + 10240 \, a^{4} c^{6} d^{2} e - 40 \, b^{9} c d e^{2} + 640 \, a b^{7} c^{2} d e^{2} - 3840 \, a^{2} b^{5} c^{3} d e^{2} + 10240 \, a^{3} b^{3} c^{4} d e^{2} - 10240 \, a^{4} b c^{5} d e^{2} + 11 \, b^{10} e^{3} - 180 \, a b^{8} c e^{3} + 1120 \, a^{2} b^{6} c^{2} e^{3} - 3200 \, a^{3} b^{4} c^{3} e^{3} + 3840 \, a^{4} b^{2} c^{4} e^{3} - 1024 \, a^{5} c^{5} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{131072 \, c^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/6881280*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(6*c^3*x*e^3 + (20*c^12*d*e^2 + 17*b*c^11*e^3)/c^9)*
x + (360*c^12*d^2*e + 920*b*c^11*d*e^2 + 251*b^2*c^10*e^3 + 252*a*c^11*e^3)/c^9)*x + (1920*c^12*d^3 + 14760*b*
c^11*d^2*e + 12120*b^2*c^10*d*e^2 + 12160*a*c^11*d*e^2 + 1035*b^3*c^9*e^3 + 6268*a*b*c^10*e^3)/c^9)*x + (23040
*b*c^11*d^3 + 57000*b^2*c^10*d^2*e + 57120*a*c^11*d^2*e + 14680*b^3*c^9*d*e^2 + 88800*a*b*c^10*d*e^2 - 5*b^4*c
^8*e^3 + 10200*a*b^2*c^9*e^3 + 10416*a^2*c^10*e^3)/c^9)*x + (46080*b^2*c^10*d^3 + 46080*a*c^11*d^3 + 35880*b^3
*c^9*d^2*e + 216480*a*b*c^10*d^2*e - 40*b^4*c^8*d*e^2 + 75360*a*b^2*c^9*d*e^2 + 76800*a^2*c^10*d*e^2 + 11*b^5*
c^7*e^3 - 120*a*b^3*c^8*e^3 + 13680*a^2*b*c^9*e^3)/c^9)*x + (122880*b^3*c^9*d^3 + 737280*a*b*c^10*d^3 - 360*b^
4*c^8*d^2*e + 782400*a*b^2*c^9*d^2*e + 792960*a^2*c^10*d^2*e + 360*b^5*c^7*d*e^2 - 4160*a*b^3*c^8*d*e^2 + 4358
40*a^2*b*c^9*d*e^2 - 99*b^6*c^6*e^3 + 1180*a*b^4*c^7*e^3 - 4560*a^2*b^2*c^8*e^3 + 6720*a^3*c^9*e^3)/c^9)*x + (
737280*a*b^2*c^9*d^3 + 737280*a^2*c^10*d^3 + 840*b^5*c^7*d^2*e - 10560*a*b^3*c^8*d^2*e + 1272960*a^2*b*c^9*d^2
*e - 840*b^6*c^6*d*e^2 + 10560*a*b^4*c^7*d*e^2 - 44160*a^2*b^2*c^8*d*e^2 + 81920*a^3*c^9*d*e^2 + 231*b^7*c^5*e
^3 - 2988*a*b^5*c^6*e^3 + 13200*a^2*b^3*c^7*e^3 - 20800*a^3*b*c^8*e^3)/c^9)*x + (2949120*a^2*b*c^9*d^3 - 4200*
b^6*c^6*d^2*e + 57120*a*b^4*c^7*d^2*e - 270720*a^2*b^2*c^8*d^2*e + 806400*a^3*c^9*d^2*e + 4200*b^7*c^5*d*e^2 -
57120*a*b^5*c^6*d*e^2 + 270720*a^2*b^3*c^7*d*e^2 - 478720*a^3*b*c^8*d*e^2 - 1155*b^8*c^4*e^3 + 16128*a*b^6*c^
5*e^3 - 80160*a^2*b^4*c^6*e^3 + 158720*a^3*b^2*c^7*e^3 - 80640*a^4*c^8*e^3)/c^9)*x + (1966080*a^3*c^9*d^3 + 12
600*b^7*c^5*d^2*e - 184800*a*b^5*c^6*d^2*e + 981120*a^2*b^3*c^7*d^2*e - 2142720*a^3*b*c^8*d^2*e - 12600*b^8*c^
4*d*e^2 + 184800*a*b^6*c^5*d*e^2 - 981120*a^2*b^4*c^6*d*e^2 + 2142720*a^3*b^2*c^7*d*e^2 - 1310720*a^4*c^8*d*e^
2 + 3465*b^9*c^3*e^3 - 52080*a*b^7*c^4*e^3 + 288288*a^2*b^5*c^5*e^3 - 687360*a^3*b^3*c^6*e^3 + 574720*a^4*b*c^
7*e^3)/c^9) + 3/131072*(40*b^8*c^2*d^2*e - 640*a*b^6*c^3*d^2*e + 3840*a^2*b^4*c^4*d^2*e - 10240*a^3*b^2*c^5*d^
2*e + 10240*a^4*c^6*d^2*e - 40*b^9*c*d*e^2 + 640*a*b^7*c^2*d*e^2 - 3840*a^2*b^5*c^3*d*e^2 + 10240*a^3*b^3*c^4*
d*e^2 - 10240*a^4*b*c^5*d*e^2 + 11*b^10*e^3 - 180*a*b^8*c*e^3 + 1120*a^2*b^6*c^2*e^3 - 3200*a^3*b^4*c^3*e^3 +
3840*a^4*b^2*c^4*e^3 - 1024*a^5*c^5*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2), x)

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