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

Optimal. Leaf size=204 \[ \frac {6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac {8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac {2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7} \]

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Rubi [A]  time = 0.09, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac {8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac {2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^7) + (2*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/(11*e^
7) + (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(13/2))/(13*e^7) - (4*c^3*d*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*
x)^(17/2))/(17*e^7)

Rule 697

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

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{e^6}-\frac {6 c^3 d (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 188, normalized size = 0.92 \begin {gather*} \frac {2 \left (\frac {3}{13} c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )-\frac {4}{11} c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )+\frac {1}{3} c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )-\frac {6}{7} c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2+\frac {1}{5} (d+e x)^{5/2} \left (a e^2+c d^2\right )^3+\frac {1}{17} c^3 (d+e x)^{17/2}-\frac {2}{5} c^3 d (d+e x)^{15/2}\right )}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/5 - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/7 + (c*(c*d^2 + a*e^2)*(
5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/3 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/11 + (3*c^2*(5*c*d^2 + a*
e^2)*(d + e*x)^(13/2))/13 - (2*c^3*d*(d + e*x)^(15/2))/5 + (c^3*(d + e*x)^(17/2))/17))/e^7

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IntegrateAlgebraic [A]  time = 0.09, size = 240, normalized size = 1.18 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (51051 a^3 e^6+153153 a^2 c d^2 e^4-218790 a^2 c d e^4 (d+e x)+85085 a^2 c e^4 (d+e x)^2+153153 a c^2 d^4 e^2-437580 a c^2 d^3 e^2 (d+e x)+510510 a c^2 d^2 e^2 (d+e x)^2-278460 a c^2 d e^2 (d+e x)^3+58905 a c^2 e^2 (d+e x)^4+51051 c^3 d^6-218790 c^3 d^5 (d+e x)+425425 c^3 d^4 (d+e x)^2-464100 c^3 d^3 (d+e x)^3+294525 c^3 d^2 (d+e x)^4-102102 c^3 d (d+e x)^5+15015 c^3 (d+e x)^6\right )}{255255 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(51051*c^3*d^6 + 153153*a*c^2*d^4*e^2 + 153153*a^2*c*d^2*e^4 + 51051*a^3*e^6 - 218790*c^3*d
^5*(d + e*x) - 437580*a*c^2*d^3*e^2*(d + e*x) - 218790*a^2*c*d*e^4*(d + e*x) + 425425*c^3*d^4*(d + e*x)^2 + 51
0510*a*c^2*d^2*e^2*(d + e*x)^2 + 85085*a^2*c*e^4*(d + e*x)^2 - 464100*c^3*d^3*(d + e*x)^3 - 278460*a*c^2*d*e^2
*(d + e*x)^3 + 294525*c^3*d^2*(d + e*x)^4 + 58905*a*c^2*e^2*(d + e*x)^4 - 102102*c^3*d*(d + e*x)^5 + 15015*c^3
*(d + e*x)^6))/(255255*e^7)

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fricas [A]  time = 0.40, size = 303, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 18018 \, c^{3} d e^{7} x^{7} + 1024 \, c^{3} d^{8} + 6528 \, a c^{2} d^{6} e^{2} + 19448 \, a^{2} c d^{4} e^{4} + 51051 \, a^{3} d^{2} e^{6} + 231 \, {\left (c^{3} d^{2} e^{6} + 255 \, a c^{2} e^{8}\right )} x^{6} - 126 \, {\left (2 \, c^{3} d^{3} e^{5} - 595 \, a c^{2} d e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} + 51 \, a c^{2} d^{2} e^{6} + 2431 \, a^{2} c e^{8}\right )} x^{4} - 10 \, {\left (32 \, c^{3} d^{5} e^{3} + 204 \, a c^{2} d^{3} e^{5} - 12155 \, a^{2} c d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} + 816 \, a c^{2} d^{4} e^{4} + 2431 \, a^{2} c d^{2} e^{6} + 17017 \, a^{3} e^{8}\right )} x^{2} - 2 \, {\left (256 \, c^{3} d^{7} e + 1632 \, a c^{2} d^{5} e^{3} + 4862 \, a^{2} c d^{3} e^{5} - 51051 \, a^{3} d e^{7}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*c^3*e^8*x^8 + 18018*c^3*d*e^7*x^7 + 1024*c^3*d^8 + 6528*a*c^2*d^6*e^2 + 19448*a^2*c*d^4*e^4 +
51051*a^3*d^2*e^6 + 231*(c^3*d^2*e^6 + 255*a*c^2*e^8)*x^6 - 126*(2*c^3*d^3*e^5 - 595*a*c^2*d*e^7)*x^5 + 35*(8*
c^3*d^4*e^4 + 51*a*c^2*d^2*e^6 + 2431*a^2*c*e^8)*x^4 - 10*(32*c^3*d^5*e^3 + 204*a*c^2*d^3*e^5 - 12155*a^2*c*d*
e^7)*x^3 + 3*(128*c^3*d^6*e^2 + 816*a*c^2*d^4*e^4 + 2431*a^2*c*d^2*e^6 + 17017*a^3*e^8)*x^2 - 2*(256*c^3*d^7*e
+ 1632*a*c^2*d^5*e^3 + 4862*a^2*c*d^3*e^5 - 51051*a^3*d*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.24, size = 834, normalized size = 4.09

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+a)^3,x, algorithm="giac")

[Out]

2/765765*(153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d^2*e^(-2) + 7293*(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^2*d^2*e^(-4) + 255*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 85
80*(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^3*d^2
*e^(-6) + 131274*(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^
2*c*d*e^(-2) + 6630*(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)*a*c^2*d*e^(-4) + 238*(429*(x*e + d)^(15/2) - 3465*
(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 2702
7*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*d*e^(-6) + 765765*sqrt(x*e + d
)*a^3*d^2 + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d + 7293*(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^2*c*e^(-2) + 765*(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)*a*c^2*e^(-4) + 7*(6435*(x*e + d)^(17/2) - 58344*
(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 -
875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^
8)*c^3*e^(-6) + 51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3)*e^(-1)

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maple [A]  time = 0.06, size = 205, normalized size = 1.00 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 c^{3} x^{6} e^{6}-12012 c^{3} d \,e^{5} x^{5}+58905 a \,c^{2} e^{6} x^{4}+9240 c^{3} d^{2} e^{4} x^{4}-42840 a \,c^{2} d \,e^{5} x^{3}-6720 c^{3} d^{3} e^{3} x^{3}+85085 a^{2} c \,e^{6} x^{2}+28560 a \,c^{2} d^{2} e^{4} x^{2}+4480 c^{3} d^{4} e^{2} x^{2}-48620 a^{2} c d \,e^{5} x -16320 a \,c^{2} d^{3} e^{3} x -2560 c^{3} d^{5} e x +51051 e^{6} a^{3}+19448 a^{2} c \,d^{2} e^{4}+6528 a \,c^{2} d^{4} e^{2}+1024 c^{3} d^{6}\right )}{255255 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(e*x+d)^(5/2)*(15015*c^3*e^6*x^6-12012*c^3*d*e^5*x^5+58905*a*c^2*e^6*x^4+9240*c^3*d^2*e^4*x^4-42840*a
*c^2*d*e^5*x^3-6720*c^3*d^3*e^3*x^3+85085*a^2*c*e^6*x^2+28560*a*c^2*d^2*e^4*x^2+4480*c^3*d^4*e^2*x^2-48620*a^2
*c*d*e^5*x-16320*a*c^2*d^3*e^3*x-2560*c^3*d^5*e*x+51051*a^3*e^6+19448*a^2*c*d^2*e^4+6528*a*c^2*d^4*e^2+1024*c^
3*d^6)/e^7

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maxima [A]  time = 1.37, size = 209, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 102102 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} d + 58905 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 92820 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 218790 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 102102*(e*x + d)^(15/2)*c^3*d + 58905*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)
^(13/2) - 92820*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)
*(e*x + d)^(9/2) - 218790*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(7/2) + 51051*(c^3*d^6 + 3*a*c^2
*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(5/2))/e^7

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mupad [B]  time = 0.37, size = 187, normalized size = 0.92 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{15/2}}{5\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^(3/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2
*d^2*e^2))/(9*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(5/2))/(5*e^7) - ((40*
c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(11/2))/(11*e^7) - (4*c^3*d*(d + e*x)^(15/2))/(5*e^7) - (12*c*d*(a*e^2 + c
*d^2)^2*(d + e*x)^(7/2))/(7*e^7)

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sympy [A]  time = 20.20, size = 564, normalized size = 2.76 \begin {gather*} a^{3} 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^{3} \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 {6 a^{2} 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 {6 a^{2} 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 {6 a 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 {6 a 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}} + \frac {2 c^{3} d \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{7}} + \frac {2 c^{3} \left (- \frac {d^{7} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {7 d^{6} \left (d + e x\right )^{\frac {5}{2}}}{5} - 3 d^{5} \left (d + e x\right )^{\frac {7}{2}} + \frac {35 d^{4} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {35 d^{3} \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {21 d^{2} \left (d + e x\right )^{\frac {13}{2}}}{13} - \frac {7 d \left (d + e x\right )^{\frac {15}{2}}}{15} + \frac {\left (d + e x\right )^{\frac {17}{2}}}{17}\right )}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 6*a**2*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
+ 6*a**2*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 + 6*a*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 + 6*a*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
+ 2*c**3*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*
x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*c**3*(-
d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 3
5*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/
e**7

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