3.24.56 \(\int (d+e x)^2 (a+b x+c x^2)^{5/2} \, dx\) [2356]

Optimal. Leaf size=323 \[ \frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \]

[Out]

-5/6144*(-4*a*c+b^2)*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/384*(32*c^2*
d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^3+9/112*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(7/2)/c^
2+1/8*e*(e*x+d)*(c*x^2+b*x+a)^(7/2)/c-5/32768*(-4*a*c+b^2)^3*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*arctanh(
1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+5/16384*(-4*a*c+b^2)^2*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*
b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5

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Rubi [A]
time = 0.27, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 654, 626, 635, 212} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^
5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(614
4*c^4) + ((32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c^3) + (9*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*
a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(32768*c^(11/2))

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 654

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

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (\frac {1}{2} \left (16 c d^2-2 e \left (\frac {7 b d}{2}+a e\right )\right )+\frac {9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (-\frac {9}{2} b e (2 c d-b e)+c \left (16 c d^2-2 e \left (\frac {7 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 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 )}{16384 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 2.29, size = 538, normalized size = 1.67 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^7 e^2-210 b^6 c e (16 d+3 e x)+28 b^5 c \left (-375 a e^2+2 c \left (60 d^2+40 d e x+9 e^2 x^2\right )\right )+8 b^4 c^2 \left (7 a e (640 d+113 e x)-2 c x \left (140 d^2+112 d e x+27 e^2 x^2\right )\right )+16 b^3 c^2 \left (2359 a^2 e^2+8 c^2 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )-4 a c \left (560 d^2+336 d e x+71 e^2 x^2\right )\right )+32 b^2 c^3 \left (-3 a^2 e (1232 d+199 e x)+12 a c x \left (56 d^2+40 d e x+9 e^2 x^2\right )+8 c^2 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )\right )+64 b c^3 \left (-663 a^3 e^2+6 a^2 c \left (308 d^2+152 d e x+29 e^2 x^2\right )+16 c^3 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )+8 a c^2 x^2 \left (546 d^2+788 d e x+307 e^2 x^2\right )\right )+128 c^4 \left (3 a^3 e (256 d+35 e x)+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )\right )\right )+105 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{688128 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^7*e^2 - 210*b^6*c*e*(16*d + 3*e*x) + 28*b^5*c*(-375*a*e^2 + 2*c*(60*d^
2 + 40*d*e*x + 9*e^2*x^2)) + 8*b^4*c^2*(7*a*e*(640*d + 113*e*x) - 2*c*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2)) +
16*b^3*c^2*(2359*a^2*e^2 + 8*c^2*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) - 4*a*c*(560*d^2 + 336*d*e*x + 71*e^2*x^2
)) + 32*b^2*c^3*(-3*a^2*e*(1232*d + 199*e*x) + 12*a*c*x*(56*d^2 + 40*d*e*x + 9*e^2*x^2) + 8*c^2*x^3*(378*d^2 +
592*d*e*x + 243*e^2*x^2)) + 64*b*c^3*(-663*a^3*e^2 + 6*a^2*c*(308*d^2 + 152*d*e*x + 29*e^2*x^2) + 16*c^3*x^4*
(140*d^2 + 232*d*e*x + 99*e^2*x^2) + 8*a*c^2*x^2*(546*d^2 + 788*d*e*x + 307*e^2*x^2)) + 128*c^4*(3*a^3*e*(256*
d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3*(182*d^2 + 288*d*e*x + 119*e^2*x^2) +
2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2))) + 105*(b^2 - 4*a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d
+ a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(688128*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(660\) vs. \(2(293)=586\).
time = 0.97, size = 661, normalized size = 2.05 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(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*d*e*(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))))))+d^2*(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((e*x+d)^2*(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. 702 vs. \(2 (301) = 602\).
time = 4.15, size = 1407, normalized size = 4.36 \begin {gather*} \left [\frac {105 \, {\left (32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d e + {\left (9 \, b^{8} - 112 \, a b^{6} c + 480 \, a^{2} b^{4} c^{2} - 768 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (57344 \, c^{8} d^{2} x^{5} + 143360 \, b c^{7} d^{2} x^{4} + 3584 \, {\left (27 \, b^{2} c^{6} + 52 \, a c^{7}\right )} d^{2} x^{3} + 1792 \, {\left (b^{3} c^{5} + 156 \, a b c^{6}\right )} d^{2} x^{2} - 448 \, {\left (5 \, b^{4} c^{4} - 48 \, a b^{2} c^{5} - 528 \, a^{2} c^{6}\right )} d^{2} x + 224 \, {\left (15 \, b^{5} c^{3} - 160 \, a b^{3} c^{4} + 528 \, a^{2} b c^{5}\right )} d^{2} + {\left (43008 \, c^{8} x^{7} + 101376 \, b c^{7} x^{6} + 945 \, b^{7} c - 10500 \, a b^{5} c^{2} + 37744 \, a^{2} b^{3} c^{3} - 42432 \, a^{3} b c^{4} + 256 \, {\left (243 \, b^{2} c^{6} + 476 \, a c^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{3} c^{5} + 1228 \, a b c^{6}\right )} x^{4} - 16 \, {\left (27 \, b^{4} c^{4} - 216 \, a b^{2} c^{5} - 6608 \, a^{2} c^{6}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{3} - 568 \, a b^{3} c^{4} + 1392 \, a^{2} b c^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{6} c^{2} - 3164 \, a b^{4} c^{3} + 9552 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} e^{2} + 32 \, {\left (3072 \, c^{8} d x^{6} + 7424 \, b c^{7} d x^{5} + 128 \, {\left (37 \, b^{2} c^{6} + 72 \, a c^{7}\right )} d x^{4} + 16 \, {\left (3 \, b^{3} c^{5} + 788 \, a b c^{6}\right )} d x^{3} - 8 \, {\left (7 \, b^{4} c^{4} - 60 \, a b^{2} c^{5} - 1152 \, a^{2} c^{6}\right )} d x^{2} + 2 \, {\left (35 \, b^{5} c^{3} - 336 \, a b^{3} c^{4} + 912 \, a^{2} b c^{5}\right )} d x - {\left (105 \, b^{6} c^{2} - 1120 \, a b^{4} c^{3} + 3696 \, a^{2} b^{2} c^{4} - 3072 \, a^{3} c^{5}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x + a}}{1376256 \, c^{6}}, \frac {105 \, {\left (32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d e + {\left (9 \, b^{8} - 112 \, a b^{6} c + 480 \, a^{2} b^{4} c^{2} - 768 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (57344 \, c^{8} d^{2} x^{5} + 143360 \, b c^{7} d^{2} x^{4} + 3584 \, {\left (27 \, b^{2} c^{6} + 52 \, a c^{7}\right )} d^{2} x^{3} + 1792 \, {\left (b^{3} c^{5} + 156 \, a b c^{6}\right )} d^{2} x^{2} - 448 \, {\left (5 \, b^{4} c^{4} - 48 \, a b^{2} c^{5} - 528 \, a^{2} c^{6}\right )} d^{2} x + 224 \, {\left (15 \, b^{5} c^{3} - 160 \, a b^{3} c^{4} + 528 \, a^{2} b c^{5}\right )} d^{2} + {\left (43008 \, c^{8} x^{7} + 101376 \, b c^{7} x^{6} + 945 \, b^{7} c - 10500 \, a b^{5} c^{2} + 37744 \, a^{2} b^{3} c^{3} - 42432 \, a^{3} b c^{4} + 256 \, {\left (243 \, b^{2} c^{6} + 476 \, a c^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{3} c^{5} + 1228 \, a b c^{6}\right )} x^{4} - 16 \, {\left (27 \, b^{4} c^{4} - 216 \, a b^{2} c^{5} - 6608 \, a^{2} c^{6}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{3} - 568 \, a b^{3} c^{4} + 1392 \, a^{2} b c^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{6} c^{2} - 3164 \, a b^{4} c^{3} + 9552 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} e^{2} + 32 \, {\left (3072 \, c^{8} d x^{6} + 7424 \, b c^{7} d x^{5} + 128 \, {\left (37 \, b^{2} c^{6} + 72 \, a c^{7}\right )} d x^{4} + 16 \, {\left (3 \, b^{3} c^{5} + 788 \, a b c^{6}\right )} d x^{3} - 8 \, {\left (7 \, b^{4} c^{4} - 60 \, a b^{2} c^{5} - 1152 \, a^{2} c^{6}\right )} d x^{2} + 2 \, {\left (35 \, b^{5} c^{3} - 336 \, a b^{3} c^{4} + 912 \, a^{2} b c^{5}\right )} d x - {\left (105 \, b^{6} c^{2} - 1120 \, a b^{4} c^{3} + 3696 \, a^{2} b^{2} c^{4} - 3072 \, a^{3} c^{5}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x + a}}{688128 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1376256*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5*c^2 + 48
*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*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*(57344*c^8
*d^2*x^5 + 143360*b*c^7*d^2*x^4 + 3584*(27*b^2*c^6 + 52*a*c^7)*d^2*x^3 + 1792*(b^3*c^5 + 156*a*b*c^6)*d^2*x^2
- 448*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2*x + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 +
(43008*c^8*x^7 + 101376*b*c^7*x^6 + 945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4 + 256*(2
43*b^2*c^6 + 476*a*c^7)*x^5 + 128*(3*b^3*c^5 + 1228*a*b*c^6)*x^4 - 16*(27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c
^6)*x^3 + 8*(63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*x^2 - 2*(315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2
*c^4 - 6720*a^3*c^5)*x)*e^2 + 32*(3072*c^8*d*x^6 + 7424*b*c^7*d*x^5 + 128*(37*b^2*c^6 + 72*a*c^7)*d*x^4 + 16*(
3*b^3*c^5 + 788*a*b*c^6)*d*x^3 - 8*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*x^2 + 2*(35*b^5*c^3 - 336*a*b^3
*c^4 + 912*a^2*b*c^5)*d*x - (105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d)*e)*sqrt(c*x^2
+ b*x + a))/c^6, 1/688128*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12
*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 2
56*a^4*c^4)*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*(
57344*c^8*d^2*x^5 + 143360*b*c^7*d^2*x^4 + 3584*(27*b^2*c^6 + 52*a*c^7)*d^2*x^3 + 1792*(b^3*c^5 + 156*a*b*c^6)
*d^2*x^2 - 448*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2*x + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^
5)*d^2 + (43008*c^8*x^7 + 101376*b*c^7*x^6 + 945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4
+ 256*(243*b^2*c^6 + 476*a*c^7)*x^5 + 128*(3*b^3*c^5 + 1228*a*b*c^6)*x^4 - 16*(27*b^4*c^4 - 216*a*b^2*c^5 - 6
608*a^2*c^6)*x^3 + 8*(63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*x^2 - 2*(315*b^6*c^2 - 3164*a*b^4*c^3 + 955
2*a^2*b^2*c^4 - 6720*a^3*c^5)*x)*e^2 + 32*(3072*c^8*d*x^6 + 7424*b*c^7*d*x^5 + 128*(37*b^2*c^6 + 72*a*c^7)*d*x
^4 + 16*(3*b^3*c^5 + 788*a*b*c^6)*d*x^3 - 8*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*x^2 + 2*(35*b^5*c^3 -
336*a*b^3*c^4 + 912*a^2*b*c^5)*d*x - (105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d)*e)*sq
rt(c*x^2 + b*x + a))/c^6]

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

[Out]

Integral((d + e*x)**2*(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. 767 vs. \(2 (301) = 602\).
time = 4.25, size = 767, normalized size = 2.37 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} x e^{2} + \frac {32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2} + 476 \, a c^{8} e^{2}}{c^{7}}\right )} x + \frac {1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 2304 \, a c^{8} d e + 3 \, b^{3} c^{6} e^{2} + 1228 \, a b c^{7} e^{2}}{c^{7}}\right )} x + \frac {6048 \, b^{2} c^{7} d^{2} + 11648 \, a c^{8} d^{2} + 96 \, b^{3} c^{6} d e + 25216 \, a b c^{7} d e - 27 \, b^{4} c^{5} e^{2} + 216 \, a b^{2} c^{6} e^{2} + 6608 \, a^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac {224 \, b^{3} c^{6} d^{2} + 34944 \, a b c^{7} d^{2} - 224 \, b^{4} c^{5} d e + 1920 \, a b^{2} c^{6} d e + 36864 \, a^{2} c^{7} d e + 63 \, b^{5} c^{4} e^{2} - 568 \, a b^{3} c^{5} e^{2} + 1392 \, a^{2} b c^{6} e^{2}}{c^{7}}\right )} x - \frac {1120 \, b^{4} c^{5} d^{2} - 10752 \, a b^{2} c^{6} d^{2} - 118272 \, a^{2} c^{7} d^{2} - 1120 \, b^{5} c^{4} d e + 10752 \, a b^{3} c^{5} d e - 29184 \, a^{2} b c^{6} d e + 315 \, b^{6} c^{3} e^{2} - 3164 \, a b^{4} c^{4} e^{2} + 9552 \, a^{2} b^{2} c^{5} e^{2} - 6720 \, a^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac {3360 \, b^{5} c^{4} d^{2} - 35840 \, a b^{3} c^{5} d^{2} + 118272 \, a^{2} b c^{6} d^{2} - 3360 \, b^{6} c^{3} d e + 35840 \, a b^{4} c^{4} d e - 118272 \, a^{2} b^{2} c^{5} d e + 98304 \, a^{3} c^{6} d e + 945 \, b^{7} c^{2} e^{2} - 10500 \, a b^{5} c^{3} e^{2} + 37744 \, a^{2} b^{3} c^{4} e^{2} - 42432 \, a^{3} b c^{5} e^{2}}{c^{7}}\right )} + \frac {5 \, {\left (32 \, b^{6} c^{2} d^{2} - 384 \, a b^{4} c^{3} d^{2} + 1536 \, a^{2} b^{2} c^{4} d^{2} - 2048 \, a^{3} c^{5} d^{2} - 32 \, b^{7} c d e + 384 \, a b^{5} c^{2} d e - 1536 \, a^{2} b^{3} c^{3} d e + 2048 \, a^{3} b c^{4} d e + 9 \, b^{8} e^{2} - 112 \, a b^{6} c e^{2} + 480 \, a^{2} b^{4} c^{2} e^{2} - 768 \, a^{3} b^{2} c^{3} e^{2} + 256 \, a^{4} c^{4} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^
9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2 + 476*a*c^8*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 2304*a*
c^8*d*e + 3*b^3*c^6*e^2 + 1228*a*b*c^7*e^2)/c^7)*x + (6048*b^2*c^7*d^2 + 11648*a*c^8*d^2 + 96*b^3*c^6*d*e + 25
216*a*b*c^7*d*e - 27*b^4*c^5*e^2 + 216*a*b^2*c^6*e^2 + 6608*a^2*c^7*e^2)/c^7)*x + (224*b^3*c^6*d^2 + 34944*a*b
*c^7*d^2 - 224*b^4*c^5*d*e + 1920*a*b^2*c^6*d*e + 36864*a^2*c^7*d*e + 63*b^5*c^4*e^2 - 568*a*b^3*c^5*e^2 + 139
2*a^2*b*c^6*e^2)/c^7)*x - (1120*b^4*c^5*d^2 - 10752*a*b^2*c^6*d^2 - 118272*a^2*c^7*d^2 - 1120*b^5*c^4*d*e + 10
752*a*b^3*c^5*d*e - 29184*a^2*b*c^6*d*e + 315*b^6*c^3*e^2 - 3164*a*b^4*c^4*e^2 + 9552*a^2*b^2*c^5*e^2 - 6720*a
^3*c^6*e^2)/c^7)*x + (3360*b^5*c^4*d^2 - 35840*a*b^3*c^5*d^2 + 118272*a^2*b*c^6*d^2 - 3360*b^6*c^3*d*e + 35840
*a*b^4*c^4*d*e - 118272*a^2*b^2*c^5*d*e + 98304*a^3*c^6*d*e + 945*b^7*c^2*e^2 - 10500*a*b^5*c^3*e^2 + 37744*a^
2*b^3*c^4*e^2 - 42432*a^3*b*c^5*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2
- 2048*a^3*c^5*d^2 - 32*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048*a^3*b*c^4*d*e + 9*b^8*e^2
- 112*a*b^6*c*e^2 + 480*a^2*b^4*c^2*e^2 - 768*a^3*b^2*c^3*e^2 + 256*a^4*c^4*e^2)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)

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

[Out]

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

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