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

Optimal. Leaf size=166 \[ \frac {2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^5}-\frac {4 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {698} \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^5}-\frac {4 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/
2))/(7*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(9/2))/(9*e^5) - (4*c*(2*c*d - b*e)*(d
+ e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{7/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{9/2}}{e^4}+\frac {c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{9/2}}{9 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 174, normalized size = 1.05 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (143 e^2 \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-26 c e \left (3 b \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-11 a e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + 143*e^2*(
63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 26*c*e*(-11*a*e*(8*d^2 - 20*d*e*
x + 35*e^2*x^2) + 3*b*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(45045*e^5)

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IntegrateAlgebraic [A]  time = 0.11, size = 229, normalized size = 1.38 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 a^2 e^4+12870 a b e^3 (d+e x)-18018 a b d e^3+18018 a c d^2 e^2-25740 a c d e^2 (d+e x)+10010 a c e^2 (d+e x)^2+9009 b^2 d^2 e^2-12870 b^2 d e^2 (d+e x)+5005 b^2 e^2 (d+e x)^2-18018 b c d^3 e+38610 b c d^2 e (d+e x)-30030 b c d e (d+e x)^2+8190 b c e (d+e x)^3+9009 c^2 d^4-25740 c^2 d^3 (d+e x)+30030 c^2 d^2 (d+e x)^2-16380 c^2 d (d+e x)^3+3465 c^2 (d+e x)^4\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(9009*c^2*d^4 - 18018*b*c*d^3*e + 9009*b^2*d^2*e^2 + 18018*a*c*d^2*e^2 - 18018*a*b*d*e^3 +
9009*a^2*e^4 - 25740*c^2*d^3*(d + e*x) + 38610*b*c*d^2*e*(d + e*x) - 12870*b^2*d*e^2*(d + e*x) - 25740*a*c*d*e
^2*(d + e*x) + 12870*a*b*e^3*(d + e*x) + 30030*c^2*d^2*(d + e*x)^2 - 30030*b*c*d*e*(d + e*x)^2 + 5005*b^2*e^2*
(d + e*x)^2 + 10010*a*c*e^2*(d + e*x)^2 - 16380*c^2*d*(d + e*x)^3 + 8190*b*c*e*(d + e*x)^3 + 3465*c^2*(d + e*x
)^4))/(45045*e^5)

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fricas [B]  time = 0.41, size = 302, normalized size = 1.82 \begin {gather*} \frac {2 \, {\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e - 5148 \, a b d^{3} e^{3} + 9009 \, a^{2} d^{2} e^{4} + 1144 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + 630 \, {\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \, {\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, {\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{4} - 10 \, {\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 1287 \, a b e^{6} - 715 \, {\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{3} + 3 \, {\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 6864 \, a b d e^{5} + 3003 \, a^{2} e^{6} + 143 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{5} e - 312 \, b c d^{4} e^{2} - 1287 \, a b d^{2} e^{4} - 9009 \, a^{2} d e^{5} + 286 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*c^2*e^6*x^6 + 384*c^2*d^6 - 1248*b*c*d^5*e - 5148*a*b*d^3*e^3 + 9009*a^2*d^2*e^4 + 1144*(b^2 + 2
*a*c)*d^4*e^2 + 630*(7*c^2*d*e^5 + 13*b*c*e^6)*x^5 + 35*(3*c^2*d^2*e^4 + 312*b*c*d*e^5 + 143*(b^2 + 2*a*c)*e^6
)*x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 1287*a*b*e^6 - 715*(b^2 + 2*a*c)*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2
- 156*b*c*d^3*e^3 + 6864*a*b*d*e^5 + 3003*a^2*e^6 + 143*(b^2 + 2*a*c)*d^2*e^4)*x^2 - 2*(96*c^2*d^5*e - 312*b*c
*d^4*e^2 - 1287*a*b*d^2*e^4 - 9009*a^2*d*e^5 + 286*(b^2 + 2*a*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.23, size = 999, normalized size = 6.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d^2*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^2*e^(-2) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*a*c*d^2*e^(-2) + 2574*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*
e + d)*d^3)*b*c*d^2*e^(-3) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(
x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*d^2*e^(-4) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*a*b*d*e^(-1) + 2574*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2
- 35*sqrt(x*e + d)*d^3)*b^2*d*e^(-2) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2
- 35*sqrt(x*e + d)*d^3)*a*c*d*e^(-2) + 572*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*
d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b*c*d*e^(-3) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*c^2*d*e^(-4) + 45045*sqrt(x*e + d)*a^2*d^2 + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d + 2574*(5*(
x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b*e^(-1) + 143*(35*(x
*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*
d^4)*b^2*e^(-2) + 286*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3
/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*c*e^(-2) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)
^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b*c*e^(-3) + 15*(231
*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e
+ d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^2*e^(-4) + 3003*(3*(x*e + d)^(5/2) - 10*
(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2)*e^(-1)

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maple [A]  time = 0.05, size = 194, normalized size = 1.17 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}+8190 b c \,e^{4} x^{3}-2520 c^{2} d \,e^{3} x^{3}+10010 a c \,e^{4} x^{2}+5005 b^{2} e^{4} x^{2}-5460 b c d \,e^{3} x^{2}+1680 c^{2} d^{2} e^{2} x^{2}+12870 a b \,e^{4} x -5720 a c d \,e^{3} x -2860 b^{2} d \,e^{3} x +3120 b c \,d^{2} e^{2} x -960 c^{2} d^{3} e x +9009 a^{2} e^{4}-5148 a b d \,e^{3}+2288 a c \,d^{2} e^{2}+1144 b^{2} d^{2} e^{2}-1248 b c \,d^{3} e +384 c^{2} d^{4}\right )}{45045 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4+8190*b*c*e^4*x^3-2520*c^2*d*e^3*x^3+10010*a*c*e^4*x^2+5005*b^2*e^4*x^2
-5460*b*c*d*e^3*x^2+1680*c^2*d^2*e^2*x^2+12870*a*b*e^4*x-5720*a*c*d*e^3*x-2860*b^2*d*e^3*x+3120*b*c*d^2*e^2*x-
960*c^2*d^3*e*x+9009*a^2*e^4-5148*a*b*d*e^3+2288*a*c*d^2*e^2+1144*b^2*d^2*e^2-1248*b*c*d^3*e+384*c^2*d^4)/e^5

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maxima [A]  time = 0.88, size = 176, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 8190 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 8190*(2*c^2*d - b*c*e)*(e*x + d)^(11/2) + 5005*(6*c^2*d^2 - 6*b*c*d*e + (
b^2 + 2*a*c)*e^2)*(e*x + d)^(9/2) - 12870*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^
(7/2) + 9009*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*(e*x + d)^(5/2))/e^5

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mupad [B]  time = 0.82, size = 148, normalized size = 0.89 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{9\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{7\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(13/2))/(13*e^5) + ((d + e*x)^(9/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(9*e^5
) + (2*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(11/2))/(11*e^5) +
(4*(b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e))/(7*e^5)

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sympy [A]  time = 24.74, size = 654, normalized size = 3.94 \begin {gather*} a^{2} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 a b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 a b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {4 a c d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {4 a c \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {4 b c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {4 b c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {2 c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 4*a*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a*b*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/
2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4*a*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)*
*(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 2*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**
(7/2)/7)/e**3 + 2*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e
*x)**(9/2)/9)/e**3 + 4*b*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 +
(d + e*x)**(9/2)/9)/e**4 + 4*b*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2
)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*
x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*
(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d
+ e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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