3.2051 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac {2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac {20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac {20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac {10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac {2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6} \]

[Out]

-2/7*(-a*e+b*d)^5*(e*x+d)^(7/2)/e^6+10/9*b*(-a*e+b*d)^4*(e*x+d)^(9/2)/e^6-20/11*b^2*(-a*e+b*d)^3*(e*x+d)^(11/2
)/e^6+20/13*b^3*(-a*e+b*d)^2*(e*x+d)^(13/2)/e^6-2/3*b^4*(-a*e+b*d)*(e*x+d)^(15/2)/e^6+2/17*b^5*(e*x+d)^(17/2)/
e^6

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Rubi [A]  time = 0.06, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 43} \[ -\frac {2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac {20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac {20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac {10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac {2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^6) - (20*b^2*(b*d - a*e
)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^6) - (2*b^4*(b*d - a*e)*(d + e*
x)^(15/2))/(3*e^6) + (2*b^5*(d + e*x)^(17/2))/(17*e^6)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^5 (d+e x)^{5/2}}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{7/2}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{9/2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{11/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{13/2}}{e^5}+\frac {b^5 (d+e x)^{15/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{7/2}}{7 e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{9/2}}{9 e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{11/2}}{11 e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{13/2}}{13 e^6}-\frac {2 b^4 (b d-a e) (d+e x)^{15/2}}{3 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 123, normalized size = 0.78 \[ \frac {2 (d+e x)^{7/2} \left (-51051 b^4 (d+e x)^4 (b d-a e)+117810 b^3 (d+e x)^3 (b d-a e)^2-139230 b^2 (d+e x)^2 (b d-a e)^3+85085 b (d+e x) (b d-a e)^4-21879 (b d-a e)^5+9009 b^5 (d+e x)^5\right )}{153153 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(-21879*(b*d - a*e)^5 + 85085*b*(b*d - a*e)^4*(d + e*x) - 139230*b^2*(b*d - a*e)^3*(d + e*x
)^2 + 117810*b^3*(b*d - a*e)^2*(d + e*x)^3 - 51051*b^4*(b*d - a*e)*(d + e*x)^4 + 9009*b^5*(d + e*x)^5))/(15315
3*e^6)

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fricas [B]  time = 1.10, size = 497, normalized size = 3.15 \[ \frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.27, size = 1698, normalized size = 10.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(1276275*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*d^3*e^(-1) + 510510*(3*(x*e + d)^(5/2) - 10*(x*e
+ d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b^2*d^3*e^(-2) + 218790*(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*b^3*d^3*e^(-3) + 12155*(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*b^4*d^3*e^(-4) + 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^5*d^3*e^(-5) + 765765*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*a^4*b*d^2*e^(-1) + 656370*(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^3*b^2*d^2*e^(-2) + 72930*(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*b^3*d^2*e^(-3) + 16575*(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 - 69
3*sqrt(x*e + d)*d^5)*a*b^4*d^2*e^(-4) + 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^5*d^2*e^(-5) + 765765*sqrt(x*e + d)*a^5*d^3 + 765765*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*d^2 + 3
28185*(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^4*b*d*e^(-1
) + 72930*(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 + 31
5*sqrt(x*e + d)*d^4)*a^3*b^2*d*e^(-2) + 33150*(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^2*b^3*d*e^(-3) + 3825*
(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*b^4*d*e^(-4) + 357*(429*(x*e + d)^(1
5/2) - 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^5*d*e^(-5) + 153153*
(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*d + 12155*(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^4*b*e^(-1) + 1105
0*(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^3*b^2*e^(-2) + 2550*(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^2*b^3*e^(-3) + 595*(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)*a*b^4*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)*b^5*e^(-5) + 2187
9*(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^5)*e^(-1)

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maple [B]  time = 0.05, size = 273, normalized size = 1.73 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 b^{5} e^{5} x^{5}+51051 a \,b^{4} e^{5} x^{4}-6006 b^{5} d \,e^{4} x^{4}+117810 a^{2} b^{3} e^{5} x^{3}-31416 a \,b^{4} d \,e^{4} x^{3}+3696 b^{5} d^{2} e^{3} x^{3}+139230 a^{3} b^{2} e^{5} x^{2}-64260 a^{2} b^{3} d \,e^{4} x^{2}+17136 a \,b^{4} d^{2} e^{3} x^{2}-2016 b^{5} d^{3} e^{2} x^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 a^{2} b^{3} d^{2} e^{3} x -7616 a \,b^{4} d^{3} e^{2} x +896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{153153 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(e*x+d)^(7/2)*(9009*b^5*e^5*x^5+51051*a*b^4*e^5*x^4-6006*b^5*d*e^4*x^4+117810*a^2*b^3*e^5*x^3-31416*a
*b^4*d*e^4*x^3+3696*b^5*d^2*e^3*x^3+139230*a^3*b^2*e^5*x^2-64260*a^2*b^3*d*e^4*x^2+17136*a*b^4*d^2*e^3*x^2-201
6*b^5*d^3*e^2*x^2+85085*a^4*b*e^5*x-61880*a^3*b^2*d*e^4*x+28560*a^2*b^3*d^2*e^3*x-7616*a*b^4*d^3*e^2*x+896*b^5
*d^4*e*x+21879*a^5*e^5-24310*a^4*b*d*e^4+17680*a^3*b^2*d^2*e^3-8160*a^2*b^3*d^3*e^2+2176*a*b^4*d^4*e-256*b^5*d
^5)/e^6

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maxima [A]  time = 0.53, size = 259, normalized size = 1.64 \[ \frac {2 \, {\left (9009 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{5} - 51051 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 117810 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 139230 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 21879 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{153153 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*(e*x + d)^(17/2)*b^5 - 51051*(b^5*d - a*b^4*e)*(e*x + d)^(15/2) + 117810*(b^5*d^2 - 2*a*b^4*d*e
+ a^2*b^3*e^2)*(e*x + d)^(13/2) - 139230*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^
(11/2) + 85085*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(9/2) - 2
1879*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(
7/2))/e^6

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mupad [B]  time = 2.05, size = 137, normalized size = 0.87 \[ \frac {2\,b^5\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^5*(d + e*x)^(17/2))/(17*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(15/2))/(15*e^6) + (2*(a*e - b*d)^5*(d
+ e*x)^(7/2))/(7*e^6) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(a*e - b*d)^2*(d + e*x)^(13
/2))/(13*e^6) + (10*b*(a*e - b*d)^4*(d + e*x)^(9/2))/(9*e^6)

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sympy [A]  time = 48.32, size = 1292, normalized size = 8.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**5*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**5*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
10*a**4*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 20*a**4*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*
(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 10*a**4*b*(-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**2 + 20*a**3*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 40*a**3*b**2*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**3 + 20*a**3*b**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**3 + 20*a**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
+ 40*a**2*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 + 20*a**2*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 10*a*b**4*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 + 20*a*b**4*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 10*a*b**4*(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 + 2*b**5*
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 + 4*b**5*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 + 2*b**5*(-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)**(1
3/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6

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