3.14.66 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=242 \[ \frac {e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{9/2}}-\frac {e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \]

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Rubi [A]  time = 0.45, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}-\frac {e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac {e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e
)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(96*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + ((24*c^2*d^2 + 7
*b^2*e^2 - 2*c*e*(7*b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(210*c^2) + ((b^2 - 4*a*c
)^3*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(9/2))

Rule 206

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

Rule 612

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 621

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 779

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 832

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)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\int (d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}-\frac {\left (\left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \sqrt {a+b x+c x^2} \, dx}{64 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac {\left (\left (b^2-4 a c\right )^3 e (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{512 c^4}\\ &=-\frac {\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac {\left (\left (b^2-4 a c\right )^3 e (2 c d-b e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{256 c^4}\\ &=-\frac {\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac {\left (b^2-4 a c\right )^3 e (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 204, normalized size = 0.84 \begin {gather*} -\frac {e \left (b^2-4 a c\right ) (b e-2 c d) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{1536 c^{9/2}}+\frac {(a+x (b+c x))^{5/2} \left (-2 c e (12 a e+7 b d+5 b e x)+7 b^2 e^2+4 c^2 d (6 d+5 e x)\right )}{210 c^2}+\frac {2}{7} (d+e x)^2 (a+x (b+c x))^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^2*(a + x*(b + c*x))^(5/2))/7 + ((a + x*(b + c*x))^(5/2)*(7*b^2*e^2 + 4*c^2*d*(6*d + 5*e*x) - 2*c*
e*(7*b*d + 12*a*e + 5*b*e*x)))/(210*c^2) - ((b^2 - 4*a*c)*e*(-2*c*d + b*e)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(
b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
x*(b + c*x)])]))/(1536*c^(9/2))

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IntegrateAlgebraic [B]  time = 2.51, size = 549, normalized size = 2.27 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-3072 a^3 c^3 e^2+3696 a^2 b^2 c^2 e^2-7392 a^2 b c^3 d e-1824 a^2 b c^3 e^2 x+10752 a^2 c^4 d^2+6720 a^2 c^4 d e x+1536 a^2 c^4 e^2 x^2-1120 a b^4 c e^2+2240 a b^3 c^2 d e+672 a b^3 c^2 e^2 x-1344 a b^2 c^3 d e x-480 a b^2 c^3 e^2 x^2+21504 a b c^4 d^2 x+25536 a b c^4 d e x^2+8896 a b c^4 e^2 x^3+21504 a c^5 d^2 x^2+31360 a c^5 d e x^3+12288 a c^5 e^2 x^4+105 b^6 e^2-210 b^5 c d e-70 b^5 c e^2 x+140 b^4 c^2 d e x+56 b^4 c^2 e^2 x^2-112 b^3 c^3 d e x^2-48 b^3 c^3 e^2 x^3+10752 b^2 c^4 d^2 x^2+15456 b^2 c^4 d e x^3+6016 b^2 c^4 e^2 x^4+21504 b c^5 d^2 x^3+34048 b c^5 d e x^4+14080 b c^5 e^2 x^5+10752 c^6 d^2 x^4+17920 c^6 d e x^5+7680 c^6 e^2 x^6\right )}{26880 c^4}+\frac {\left (-64 a^3 b c^3 e^2+128 a^3 c^4 d e+48 a^2 b^3 c^2 e^2-96 a^2 b^2 c^3 d e-12 a b^5 c e^2+24 a b^4 c^2 d e+b^7 e^2-2 b^6 c d e\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{512 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(10752*a^2*c^4*d^2 - 210*b^5*c*d*e + 2240*a*b^3*c^2*d*e - 7392*a^2*b*c^3*d*e + 105*b^6*
e^2 - 1120*a*b^4*c*e^2 + 3696*a^2*b^2*c^2*e^2 - 3072*a^3*c^3*e^2 + 21504*a*b*c^4*d^2*x + 140*b^4*c^2*d*e*x - 1
344*a*b^2*c^3*d*e*x + 6720*a^2*c^4*d*e*x - 70*b^5*c*e^2*x + 672*a*b^3*c^2*e^2*x - 1824*a^2*b*c^3*e^2*x + 10752
*b^2*c^4*d^2*x^2 + 21504*a*c^5*d^2*x^2 - 112*b^3*c^3*d*e*x^2 + 25536*a*b*c^4*d*e*x^2 + 56*b^4*c^2*e^2*x^2 - 48
0*a*b^2*c^3*e^2*x^2 + 1536*a^2*c^4*e^2*x^2 + 21504*b*c^5*d^2*x^3 + 15456*b^2*c^4*d*e*x^3 + 31360*a*c^5*d*e*x^3
- 48*b^3*c^3*e^2*x^3 + 8896*a*b*c^4*e^2*x^3 + 10752*c^6*d^2*x^4 + 34048*b*c^5*d*e*x^4 + 6016*b^2*c^4*e^2*x^4
+ 12288*a*c^5*e^2*x^4 + 17920*c^6*d*e*x^5 + 14080*b*c^5*e^2*x^5 + 7680*c^6*e^2*x^6))/(26880*c^4) + ((-2*b^6*c*
d*e + 24*a*b^4*c^2*d*e - 96*a^2*b^2*c^3*d*e + 128*a^3*c^4*d*e + b^7*e^2 - 12*a*b^5*c*e^2 + 48*a^2*b^3*c^2*e^2
- 64*a^3*b*c^3*e^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(512*c^(9/2))

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fricas [B]  time = 0.55, size = 1015, normalized size = 4.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/107520*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^
2 - 64*a^3*b*c^3)*e^2)*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*(7680*c^7*e^2*x^6 + 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c^6*e^2)*x^5 + 128*(84*c^7*d^2 + 26
6*b*c^6*d*e + (47*b^2*c^5 + 96*a*c^6)*e^2)*x^4 + 16*(1344*b*c^6*d^2 + 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3
*c^4 - 556*a*b*c^5)*e^2)*x^3 - 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d*e + (105*b^6*c - 1120*a*b^4*c
^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e^2 + 8*(1344*(b^2*c^5 + 2*a*c^6)*d^2 - 14*(b^3*c^4 - 228*a*b*c^5)*d*e +
(7*b^4*c^3 - 60*a*b^2*c^4 + 192*a^2*c^5)*e^2)*x^2 + 2*(10752*a*b*c^5*d^2 + 14*(5*b^4*c^3 - 48*a*b^2*c^4 + 240
*a^2*c^5)*d*e - (35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/53760*(105
*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c
^3)*e^2)*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*(7680*c^7
*e^2*x^6 + 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c^6*e^2)*x^5 + 128*(84*c^7*d^2 + 266*b*c^6*d*e + (47*b^
2*c^5 + 96*a*c^6)*e^2)*x^4 + 16*(1344*b*c^6*d^2 + 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3*c^4 - 556*a*b*c^5)*
e^2)*x^3 - 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d*e + (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^
3 - 3072*a^3*c^4)*e^2 + 8*(1344*(b^2*c^5 + 2*a*c^6)*d^2 - 14*(b^3*c^4 - 228*a*b*c^5)*d*e + (7*b^4*c^3 - 60*a*b
^2*c^4 + 192*a^2*c^5)*e^2)*x^2 + 2*(10752*a*b*c^5*d^2 + 14*(5*b^4*c^3 - 48*a*b^2*c^4 + 240*a^2*c^5)*d*e - (35*
b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5]

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giac [B]  time = 0.25, size = 534, normalized size = 2.21 \begin {gather*} \frac {1}{26880} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (6 \, c^{2} x e^{2} + \frac {14 \, c^{8} d e + 11 \, b c^{7} e^{2}}{c^{6}}\right )} x + \frac {84 \, c^{8} d^{2} + 266 \, b c^{7} d e + 47 \, b^{2} c^{6} e^{2} + 96 \, a c^{7} e^{2}}{c^{6}}\right )} x + \frac {1344 \, b c^{7} d^{2} + 966 \, b^{2} c^{6} d e + 1960 \, a c^{7} d e - 3 \, b^{3} c^{5} e^{2} + 556 \, a b c^{6} e^{2}}{c^{6}}\right )} x + \frac {1344 \, b^{2} c^{6} d^{2} + 2688 \, a c^{7} d^{2} - 14 \, b^{3} c^{5} d e + 3192 \, a b c^{6} d e + 7 \, b^{4} c^{4} e^{2} - 60 \, a b^{2} c^{5} e^{2} + 192 \, a^{2} c^{6} e^{2}}{c^{6}}\right )} x + \frac {10752 \, a b c^{6} d^{2} + 70 \, b^{4} c^{4} d e - 672 \, a b^{2} c^{5} d e + 3360 \, a^{2} c^{6} d e - 35 \, b^{5} c^{3} e^{2} + 336 \, a b^{3} c^{4} e^{2} - 912 \, a^{2} b c^{5} e^{2}}{c^{6}}\right )} x + \frac {10752 \, a^{2} c^{6} d^{2} - 210 \, b^{5} c^{3} d e + 2240 \, a b^{3} c^{4} d e - 7392 \, a^{2} b c^{5} d e + 105 \, b^{6} c^{2} e^{2} - 1120 \, a b^{4} c^{3} e^{2} + 3696 \, a^{2} b^{2} c^{4} e^{2} - 3072 \, a^{3} c^{5} e^{2}}{c^{6}}\right )} - \frac {{\left (2 \, b^{6} c d e - 24 \, a b^{4} c^{2} d e + 96 \, a^{2} b^{2} c^{3} d e - 128 \, a^{3} c^{4} d e - b^{7} e^{2} + 12 \, a b^{5} c e^{2} - 48 \, a^{2} b^{3} c^{2} e^{2} + 64 \, a^{3} b c^{3} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{512 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26880*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(6*c^2*x*e^2 + (14*c^8*d*e + 11*b*c^7*e^2)/c^6)*x + (84*c^8*d^2
+ 266*b*c^7*d*e + 47*b^2*c^6*e^2 + 96*a*c^7*e^2)/c^6)*x + (1344*b*c^7*d^2 + 966*b^2*c^6*d*e + 1960*a*c^7*d*e -
3*b^3*c^5*e^2 + 556*a*b*c^6*e^2)/c^6)*x + (1344*b^2*c^6*d^2 + 2688*a*c^7*d^2 - 14*b^3*c^5*d*e + 3192*a*b*c^6*
d*e + 7*b^4*c^4*e^2 - 60*a*b^2*c^5*e^2 + 192*a^2*c^6*e^2)/c^6)*x + (10752*a*b*c^6*d^2 + 70*b^4*c^4*d*e - 672*a
*b^2*c^5*d*e + 3360*a^2*c^6*d*e - 35*b^5*c^3*e^2 + 336*a*b^3*c^4*e^2 - 912*a^2*b*c^5*e^2)/c^6)*x + (10752*a^2*
c^6*d^2 - 210*b^5*c^3*d*e + 2240*a*b^3*c^4*d*e - 7392*a^2*b*c^5*d*e + 105*b^6*c^2*e^2 - 1120*a*b^4*c^3*e^2 + 3
696*a^2*b^2*c^4*e^2 - 3072*a^3*c^5*e^2)/c^6) - 1/512*(2*b^6*c*d*e - 24*a*b^4*c^2*d*e + 96*a^2*b^2*c^3*d*e - 12
8*a^3*c^4*d*e - b^7*e^2 + 12*a*b^5*c*e^2 - 48*a^2*b^3*c^2*e^2 + 64*a^3*b*c^3*e^2)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)

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maple [B]  time = 0.08, size = 895, normalized size = 3.70 \begin {gather*} \frac {a^{3} b \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}-\frac {a^{3} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 \sqrt {c}}-\frac {3 a^{2} b^{3} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}+\frac {3 a^{2} b^{2} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 a \,b^{5} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}-\frac {3 a \,b^{4} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}-\frac {b^{7} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {9}{2}}}+\frac {b^{6} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b \,e^{2} x}{8 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} d e x}{4}-\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} e^{2} x}{16 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{2} d e x}{8 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{5} e^{2} x}{128 c^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b^{4} d e x}{64 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} e^{2}}{16 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b d e}{8 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{4} e^{2}}{32 c^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} d e}{16 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,e^{2} x}{12 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a d e x}{6}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{6} e^{2}}{256 c^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b^{5} d e}{128 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} e^{2} x}{48 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} d e x}{24 c}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} e^{2} x^{2}}{7}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} e^{2}}{24 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b d e}{12 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} e^{2}}{96 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} d e}{48 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,e^{2} x}{21 c}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} d e x}{3}-\frac {4 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,e^{2}}{35 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} e^{2}}{30 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b d e}{15 c}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} d^{2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/c*e^2*b*a^2*(c*x^2+b*x+a)^(1/2)*x+1/12/c*e^2*b*a*x*(c*x^2+b*x+a)^(3/2)-1/16/c^2*e^2*b^3*(c*x^2+b*x+a)^(1/2
)*x*a-1/12*a/c*(c*x^2+b*x+a)^(3/2)*b*d*e-1/8*a^2/c*(c*x^2+b*x+a)^(1/2)*b*d*e+1/8*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a
*d*e-1/512/c^(9/2)*e^2*b^7*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/96/c^3*e^2*b^4*(c*x^2+b*x+a)^(3/2)+1/
256/c^4*e^2*b^6*(c*x^2+b*x+a)^(1/2)-4/35/c*e^2*a*(c*x^2+b*x+a)^(5/2)+1/30/c^2*e^2*b^2*(c*x^2+b*x+a)^(5/2)+2/3*
x*(c*x^2+b*x+a)^(5/2)*d*e+2/7*e^2*x^2*(c*x^2+b*x+a)^(5/2)-1/4*a^3/c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*d*e+2/5*(c*x^2+b*x+a)^(5/2)*d^2+1/256*b^6/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e-1/6*
a*x*(c*x^2+b*x+a)^(3/2)*d*e-1/21/c*e^2*b*x*(c*x^2+b*x+a)^(5/2)-1/48/c^2*e^2*b^3*x*(c*x^2+b*x+a)^(3/2)+1/128/c^
3*e^2*b^5*(c*x^2+b*x+a)^(1/2)*x-1/4*a^2*(c*x^2+b*x+a)^(1/2)*x*d*e+1/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a*d*e+1/24*
b^2/c*x*(c*x^2+b*x+a)^(3/2)*d*e+1/8/c^(3/2)*e^2*b*a^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/32/c^(5/2)
*e^2*b^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/128/c^(7/2)*e^2*b^5*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a-1/15*b/c*(c*x^2+b*x+a)^(5/2)*d*e+1/48*b^3/c^2*(c*x^2+b*x+a)^(3/2)*d*e-1/128*b^5/c^3*(c*x^2+b*x+
a)^(1/2)*d*e+1/24/c^2*e^2*b^2*a*(c*x^2+b*x+a)^(3/2)+1/16/c^2*e^2*b^2*a^2*(c*x^2+b*x+a)^(1/2)-1/32/c^3*e^2*b^4*
(c*x^2+b*x+a)^(1/2)*a-1/64*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*d*e+3/16*b^2/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a^2*d*e-3/64*b^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e

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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)^2*(c*x^2+b*x+a)^(3/2),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 details)Is 4*a*c-b^2 positive, negative or zero?

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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 )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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