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3.4.32 \(\int (d+e x)^{5/2} (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=248 \[ \frac {2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac {6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}-\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]

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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 {2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac {6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}-\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac {2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 -
 10*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(15/2))/(
5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*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)^{5/2} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 (d+e x)^{5/2}}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{7/2}}{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)^{9/2}}{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)^{11/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{15/2}}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 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)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 206, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (969969 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-373065 (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 )+1322685 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 )-855855 c^2 (d+e x)^5 (2 c d-b e)+692835 d^3 (c d-b e)^3-1616615 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+255255 c^3 (d+e x)^6\right )}{4849845 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(692835*d^3*(c*d - b*e)^3 - 1616615*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1322685*d*(
c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 373065*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2
*e^2)*(d + e*x)^3 + 969969*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 855855*c^2*(2*c*d - b*e)*(d + e*x
)^5 + 255255*c^3*(d + e*x)^6))/(4849845*e^7)

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IntegrateAlgebraic [A]  time = 0.12, size = 335, normalized size = 1.35 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-692835 b^3 d^3 e^3+1616615 b^3 d^2 e^3 (d+e x)-1322685 b^3 d e^3 (d+e x)^2+373065 b^3 e^3 (d+e x)^3+2078505 b^2 c d^4 e^2-6466460 b^2 c d^3 e^2 (d+e x)+7936110 b^2 c d^2 e^2 (d+e x)^2-4476780 b^2 c d e^2 (d+e x)^3+969969 b^2 c e^2 (d+e x)^4-2078505 b c^2 d^5 e+8083075 b c^2 d^4 e (d+e x)-13226850 b c^2 d^3 e (d+e x)^2+11191950 b c^2 d^2 e (d+e x)^3-4849845 b c^2 d e (d+e x)^4+855855 b c^2 e (d+e x)^5+692835 c^3 d^6-3233230 c^3 d^5 (d+e x)+6613425 c^3 d^4 (d+e x)^2-7461300 c^3 d^3 (d+e x)^3+4849845 c^3 d^2 (d+e x)^4-1711710 c^3 d (d+e x)^5+255255 c^3 (d+e x)^6\right )}{4849845 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(692835*c^3*d^6 - 2078505*b*c^2*d^5*e + 2078505*b^2*c*d^4*e^2 - 692835*b^3*d^3*e^3 - 323323
0*c^3*d^5*(d + e*x) + 8083075*b*c^2*d^4*e*(d + e*x) - 6466460*b^2*c*d^3*e^2*(d + e*x) + 1616615*b^3*d^2*e^3*(d
 + e*x) + 6613425*c^3*d^4*(d + e*x)^2 - 13226850*b*c^2*d^3*e*(d + e*x)^2 + 7936110*b^2*c*d^2*e^2*(d + e*x)^2 -
 1322685*b^3*d*e^3*(d + e*x)^2 - 7461300*c^3*d^3*(d + e*x)^3 + 11191950*b*c^2*d^2*e*(d + e*x)^3 - 4476780*b^2*
c*d*e^2*(d + e*x)^3 + 373065*b^3*e^3*(d + e*x)^3 + 4849845*c^3*d^2*(d + e*x)^4 - 4849845*b*c^2*d*e*(d + e*x)^4
 + 969969*b^2*c*e^2*(d + e*x)^4 - 1711710*c^3*d*(d + e*x)^5 + 855855*b*c^2*e*(d + e*x)^5 + 255255*c^3*(d + e*x
)^6))/(4849845*e^7)

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fricas [A]  time = 0.41, size = 426, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e + 41344 \, b^{2} c d^{7} e^{2} - 25840 \, b^{3} d^{6} e^{3} + 45045 \, {\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \, {\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, b^{2} c e^{9}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, b^{2} c d e^{8} + 1615 \, b^{3} e^{9}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, b^{2} c d^{2} e^{7} - 14535 \, b^{3} d e^{8}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 323 \, b^{2} c d^{3} e^{6} + 17119 \, b^{3} d^{2} e^{7}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} + 2584 \, b^{2} c d^{4} e^{5} - 1615 \, b^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (320 \, c^{3} d^{7} e^{2} - 1520 \, b c^{2} d^{6} e^{3} + 2584 \, b^{2} c d^{5} e^{4} - 1615 \, b^{3} d^{4} e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{8} e - 1520 \, b c^{2} d^{7} e^{2} + 2584 \, b^{2} c d^{6} e^{3} - 1615 \, b^{3} d^{5} e^{4}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e + 41344*b^2*c*d^7*e^2 - 25840*b^3*d^6*e^3 + 4
5045*(13*c^3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*b^2*c*e^9)*x^7 + 231*(5
*c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*b^2*c*d*e^8 + 1615*b^3*e^9)*x^6 - 63*(20*c^3*d^4*e^5 - 95*b*c^2*d^3*
e^6 - 22933*b^2*c*d^2*e^7 - 14535*b^3*d*e^8)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 323*b^2*c*d^3*e^6
+ 17119*b^3*d^2*e^7)*x^4 - 5*(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 + 2584*b^2*c*d^4*e^5 - 1615*b^3*d^3*e^6)*x^
3 + 6*(320*c^3*d^7*e^2 - 1520*b*c^2*d^6*e^3 + 2584*b^2*c*d^5*e^4 - 1615*b^3*d^4*e^5)*x^2 - 8*(320*c^3*d^8*e -
1520*b*c^2*d^7*e^2 + 2584*b^2*c*d^6*e^3 - 1615*b^3*d^5*e^4)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.26, size = 1545, normalized size = 6.23

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/4849845*(138567*(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^3*e^(-3) + 46189*(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^3*e^(-4) + 20995*(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^3*e^
(-5) + 1615*(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^3*e^(-6) + 46189*(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^2*e^(-3) + 62985*(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^2*e^(-4) + 14535*(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*c^2*d^2*e^(-5) + 2261*(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 - 270
27*(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^2*e^(-6) + 20995*(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*d*e^(-3) + 14535*(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*d*e^(-4) + 6783*(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*d*e^(-5) + 133*(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*d*e^(-6) + 1615*(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^3*e^(-3) + 2261*(429*(x*e + d)^(15/2) - 3
465*(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^2*c*e^(-4) + 133*(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 + 109
395*sqrt(x*e + d)*d^8)*b*c^2*e^(-5) + 21*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d
)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 277
1340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9
)*c^3*e^(-6))*e^(-1)

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maple [A]  time = 0.05, size = 286, normalized size = 1.15 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-255255 c^{3} x^{6} e^{6}-855855 b \,c^{2} e^{6} x^{5}+180180 c^{3} d \,e^{5} x^{5}-969969 b^{2} c \,e^{6} x^{4}+570570 b \,c^{2} d \,e^{5} x^{4}-120120 c^{3} d^{2} e^{4} x^{4}-373065 b^{3} e^{6} x^{3}+596904 b^{2} c d \,e^{5} x^{3}-351120 b \,c^{2} d^{2} e^{4} x^{3}+73920 c^{3} d^{3} e^{3} x^{3}+203490 b^{3} d \,e^{5} x^{2}-325584 b^{2} c \,d^{2} e^{4} x^{2}+191520 b \,c^{2} d^{3} e^{3} x^{2}-40320 c^{3} d^{4} e^{2} x^{2}-90440 b^{3} d^{2} e^{4} x +144704 b^{2} c \,d^{3} e^{3} x -85120 b \,c^{2} d^{4} e^{2} x +17920 c^{3} d^{5} e x +25840 b^{3} d^{3} e^{3}-41344 b^{2} c \,d^{4} e^{2}+24320 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{4849845 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+b*x)^3,x)

[Out]

-2/4849845*(e*x+d)^(7/2)*(-255255*c^3*e^6*x^6-855855*b*c^2*e^6*x^5+180180*c^3*d*e^5*x^5-969969*b^2*c*e^6*x^4+5
70570*b*c^2*d*e^5*x^4-120120*c^3*d^2*e^4*x^4-373065*b^3*e^6*x^3+596904*b^2*c*d*e^5*x^3-351120*b*c^2*d^2*e^4*x^
3+73920*c^3*d^3*e^3*x^3+203490*b^3*d*e^5*x^2-325584*b^2*c*d^2*e^4*x^2+191520*b*c^2*d^3*e^3*x^2-40320*c^3*d^4*e
^2*x^2-90440*b^3*d^2*e^4*x+144704*b^2*c*d^3*e^3*x-85120*b*c^2*d^4*e^2*x+17920*c^3*d^5*e*x+25840*b^3*d^3*e^3-41
344*b^2*c*d^4*e^2+24320*b*c^2*d^5*e-5120*c^3*d^6)/e^7

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maxima [A]  time = 1.39, size = 271, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 373065 \, {\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 {13}{2}} + 1322685 \, {\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 {11}{2}} - 1616615 \, {\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 {9}{2}} + 692835 \, {\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 {7}{2}}\right )}}{4849845 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b
*c^2*d*e + b^2*c*e^2)*(e*x + d)^(15/2) - 373065*(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)^(13/2) + 1322685*(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)^(11/2) - 1616615*(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)^(9/2) + 692835*(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)^(7/2))/e^7

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mupad [B]  time = 0.20, size = 239, normalized size = 0.96 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\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 )}{13\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\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 )}{11\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{3\,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)^(5/2),x)

[Out]

((d + e*x)^(13/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(13*e^7) + (2*c^3*(d + e*x)^(19/
2))/(19*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(17/2))/(17*e^7) + ((d + e*x)^(15/2)*(30*c^3*d^2 + 6*b^2*c*e^
2 - 30*b*c^2*d*e))/(15*e^7) + ((d + e*x)^(11/2)*(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)
)/(11*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(7/2))/(7*e^7) + (2*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(9/2
))/(3*e^7)

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sympy [B]  time = 43.85, size = 1207, normalized size = 4.87

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(c*x**2+b*x)**3,x)

[Out]

2*b**3*d**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**4 + 4*b**3*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**4 + 2*b**3*(-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**4 + 6
*b**2*c*d**2*(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 + 12*b**2*c*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**5 + 6*
b**2*c*(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**5 + 6*b*c**2*d**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**6 + 12*b*c**2*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**6 + 6*b*c**2*(-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**6 + 2*c**3*d**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**7 + 4*c**3*d*(-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 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 -
 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/
11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2
)/19)/e**7

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