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

Optimal. Leaf size=248 \[ \frac {6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac {2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac {2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}-\frac {6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac {2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}-\frac {6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac {2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(5/2))/(5*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) +
(2*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13
*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7)

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 (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 (d+e x)^{3/2}}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{5/2}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 (d+e x)^{5/2}}{5 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac {2 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 206, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (58905 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-23205 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+85085 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-51051 c^2 (d+e x)^5 (2 c d-b e)+51051 d^3 (c d-b e)^3-109395 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+15015 c^3 (d+e x)^6\right )}{255255 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(51051*d^3*(c*d - b*e)^3 - 109395*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 85085*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 23205*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)
*(d + e*x)^3 + 58905*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 51051*c^2*(2*c*d - b*e)*(d + e*x)^5 + 1
5015*c^3*(d + e*x)^6))/(255255*e^7)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.11, size = 335, normalized size = 1.35 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-51051 b^3 d^3 e^3+109395 b^3 d^2 e^3 (d+e x)-85085 b^3 d e^3 (d+e x)^2+23205 b^3 e^3 (d+e x)^3+153153 b^2 c d^4 e^2-437580 b^2 c d^3 e^2 (d+e x)+510510 b^2 c d^2 e^2 (d+e x)^2-278460 b^2 c d e^2 (d+e x)^3+58905 b^2 c e^2 (d+e x)^4-153153 b c^2 d^5 e+546975 b c^2 d^4 e (d+e x)-850850 b c^2 d^3 e (d+e x)^2+696150 b c^2 d^2 e (d+e x)^3-294525 b c^2 d e (d+e x)^4+51051 b c^2 e (d+e x)^5+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)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(5/2)*(51051*c^3*d^6 - 153153*b*c^2*d^5*e + 153153*b^2*c*d^4*e^2 - 51051*b^3*d^3*e^3 - 218790*c^3
*d^5*(d + e*x) + 546975*b*c^2*d^4*e*(d + e*x) - 437580*b^2*c*d^3*e^2*(d + e*x) + 109395*b^3*d^2*e^3*(d + e*x)
+ 425425*c^3*d^4*(d + e*x)^2 - 850850*b*c^2*d^3*e*(d + e*x)^2 + 510510*b^2*c*d^2*e^2*(d + e*x)^2 - 85085*b^3*d
*e^3*(d + e*x)^2 - 464100*c^3*d^3*(d + e*x)^3 + 696150*b*c^2*d^2*e*(d + e*x)^3 - 278460*b^2*c*d*e^2*(d + e*x)^
3 + 23205*b^3*e^3*(d + e*x)^3 + 294525*c^3*d^2*(d + e*x)^4 - 294525*b*c^2*d*e*(d + e*x)^4 + 58905*b^2*c*e^2*(d
+ e*x)^4 - 102102*c^3*d*(d + e*x)^5 + 51051*b*c^2*e*(d + e*x)^5 + 15015*c^3*(d + e*x)^6))/(255255*e^7)

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 373, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e + 6528 \, b^{2} c d^{6} e^{2} - 3536 \, b^{3} d^{5} e^{3} + 3003 \, {\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \, {\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, b^{2} c e^{8}\right )} x^{6} - 21 \, {\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, b^{2} c d e^{7} - 1105 \, b^{3} e^{8}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, b^{2} c d^{2} e^{6} + 884 \, b^{3} d e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} + 408 \, b^{2} c d^{3} e^{5} - 221 \, b^{3} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} + 408 \, b^{2} c d^{4} e^{4} - 221 \, b^{3} d^{3} e^{5}\right )} x^{2} - 8 \, {\left (64 \, c^{3} d^{7} e - 272 \, b c^{2} d^{6} e^{2} + 408 \, b^{2} c d^{5} e^{3} - 221 \, b^{3} d^{4} e^{4}\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+b*x)^3,x, algorithm="fricas")

[Out]

2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e + 6528*b^2*c*d^6*e^2 - 3536*b^3*d^5*e^3 + 3003*(
6*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 + 231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*b^2*c*e^8)*x^6 - 21*(12*c^3*d^3*e^5
- 51*b*c^2*d^2*e^6 - 3570*b^2*c*d*e^7 - 1105*b^3*e^8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^3*e^5 + 51*b^2*c*d
^2*e^6 + 884*b^3*d*e^7)*x^4 - 5*(64*c^3*d^5*e^3 - 272*b*c^2*d^4*e^4 + 408*b^2*c*d^3*e^5 - 221*b^3*d^2*e^6)*x^3
+ 6*(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 + 408*b^2*c*d^4*e^4 - 221*b^3*d^3*e^5)*x^2 - 8*(64*c^3*d^7*e - 272*b*
c^2*d^6*e^2 + 408*b^2*c*d^5*e^3 - 221*b^3*d^4*e^4)*x)*sqrt(e*x + d)/e^7

________________________________________________________________________________________

giac [B]  time = 0.22, size = 1077, normalized size = 4.34

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)^3,x, algorithm="giac")

[Out]

2/765765*(21879*(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^3
*d^2*e^(-3) + 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)*b^2*c*d^2*e^(-4) + 3315*(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^2*d^2*e^(-5)
+ 255*(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^3*d^2*e^(-6) + 4862*(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^3*d*e^(-3) + 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)*b^2*c*d*e^(-4) + 1530*(231*(x*e + d)^(13/2) - 16
38*(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)*b*c^2*d*e^(-5) + 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 - 27027*(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) + 1105*(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^3*e^(-3) + 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)*b^2*c*e^(
-4) + 357*(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 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*
d^7)*b*c^2*e^(-5) + 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))*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 286, normalized size = 1.15 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (-15015 c^{3} x^{6} e^{6}-51051 b \,c^{2} e^{6} x^{5}+12012 c^{3} d \,e^{5} x^{5}-58905 b^{2} c \,e^{6} x^{4}+39270 b \,c^{2} d \,e^{5} x^{4}-9240 c^{3} d^{2} e^{4} x^{4}-23205 b^{3} e^{6} x^{3}+42840 b^{2} c d \,e^{5} x^{3}-28560 b \,c^{2} d^{2} e^{4} x^{3}+6720 c^{3} d^{3} e^{3} x^{3}+15470 b^{3} d \,e^{5} x^{2}-28560 b^{2} c \,d^{2} e^{4} x^{2}+19040 b \,c^{2} d^{3} e^{3} x^{2}-4480 c^{3} d^{4} e^{2} x^{2}-8840 b^{3} d^{2} e^{4} x +16320 b^{2} c \,d^{3} e^{3} x -10880 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +3536 b^{3} d^{3} e^{3}-6528 b^{2} c \,d^{4} e^{2}+4352 b \,c^{2} d^{5} e -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+b*x)^3,x)

[Out]

-2/255255*(e*x+d)^(5/2)*(-15015*c^3*e^6*x^6-51051*b*c^2*e^6*x^5+12012*c^3*d*e^5*x^5-58905*b^2*c*e^6*x^4+39270*
b*c^2*d*e^5*x^4-9240*c^3*d^2*e^4*x^4-23205*b^3*e^6*x^3+42840*b^2*c*d*e^5*x^3-28560*b*c^2*d^2*e^4*x^3+6720*c^3*
d^3*e^3*x^3+15470*b^3*d*e^5*x^2-28560*b^2*c*d^2*e^4*x^2+19040*b*c^2*d^3*e^3*x^2-4480*c^3*d^4*e^2*x^2-8840*b^3*
d^2*e^4*x+16320*b^2*c*d^3*e^3*x-10880*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+3536*b^3*d^3*e^3-6528*b^2*c*d^4*e^2+435
2*b*c^2*d^5*e-1024*c^3*d^6)/e^7

________________________________________________________________________________________

maxima [A]  time = 1.42, size = 271, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 51051 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 23205 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\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+b*x)^3,x, algorithm="maxima")

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2
*d*e + b^2*c*e^2)*(e*x + d)^(13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^
(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(9/2) - 109395*(2*c^3*d^5
- 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^
4*e^2 - b^3*d^3*e^3)*(e*x + d)^(5/2))/e^7

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 239, normalized size = 0.96 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{9\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(11/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(11*e^7) + (2*c^3*(d + e*x)^(17/
2))/(17*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(15/2))/(15*e^7) + ((d + e*x)^(13/2)*(30*c^3*d^2 + 6*b^2*c*e^
2 - 30*b*c^2*d*e))/(13*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e))
/(9*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(5/2))/(5*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(7/2))
/(7*e^7)

________________________________________________________________________________________

sympy [B]  time = 27.32, size = 738, normalized size = 2.98 \begin {gather*} \frac {2 b^{3} 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 {2 b^{3} \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 {6 b^{2} c 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 b^{2} c \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 {6 b c^{2} d \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^{6}} + \frac {6 b c^{2} \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^{6}} + \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+b*x)**3,x)

[Out]

2*b**3*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 + 2*b**3*(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 + 6*b**2*c*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*b**2*c*(-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 + 6*b*c**2*d*(-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**6 + 6*b*c**2*
(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**6 + 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 - 35*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

________________________________________________________________________________________