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

Optimal. Leaf size=376 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{3 e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^6}{5 e^7 (a+b x)}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{5 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^7 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{5 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^7 (a+b x)}+\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{3 e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^6}{5 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) - (12*b*(b*d - a*e)^5*(d + e
*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(9/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))
- (4*b^5*(b*d - a*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(17/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^{3/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^{3/2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6 (d+e x)^{3/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{5/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{7/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{9/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13/2}}{e^6}+\frac {b^6 (d+e x)^{15/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^6 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {12 b (b d-a e)^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}-\frac {4 b^5 (b d-a e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 163, normalized size = 0.43 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (-102102 b^5 (d+e x)^5 (b d-a e)+294525 b^4 (d+e x)^4 (b d-a e)^2-464100 b^3 (d+e x)^3 (b d-a e)^3+425425 b^2 (d+e x)^2 (b d-a e)^4-218790 b (d+e x) (b d-a e)^5+51051 (b d-a e)^6+15015 b^6 (d+e x)^6\right )}{255255 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(51051*(b*d - a*e)^6 - 218790*b*(b*d - a*e)^5*(d + e*x) + 425425*b^2*(b*d
- a*e)^4*(d + e*x)^2 - 464100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 294525*b^4*(b*d - a*e)^2*(d + e*x)^4 - 102102*b
^5*(b*d - a*e)*(d + e*x)^5 + 15015*b^6*(d + e*x)^6))/(255255*e^7*(a + b*x))

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IntegrateAlgebraic [A]  time = 50.75, size = 466, normalized size = 1.24 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (51051 a^6 e^6+218790 a^5 b e^5 (d+e x)-306306 a^5 b d e^5+765765 a^4 b^2 d^2 e^4+425425 a^4 b^2 e^4 (d+e x)^2-1093950 a^4 b^2 d e^4 (d+e x)-1021020 a^3 b^3 d^3 e^3+2187900 a^3 b^3 d^2 e^3 (d+e x)+464100 a^3 b^3 e^3 (d+e x)^3-1701700 a^3 b^3 d e^3 (d+e x)^2+765765 a^2 b^4 d^4 e^2-2187900 a^2 b^4 d^3 e^2 (d+e x)+2552550 a^2 b^4 d^2 e^2 (d+e x)^2+294525 a^2 b^4 e^2 (d+e x)^4-1392300 a^2 b^4 d e^2 (d+e x)^3-306306 a b^5 d^5 e+1093950 a b^5 d^4 e (d+e x)-1701700 a b^5 d^3 e (d+e x)^2+1392300 a b^5 d^2 e (d+e x)^3+102102 a b^5 e (d+e x)^5-589050 a b^5 d e (d+e x)^4+51051 b^6 d^6-218790 b^6 d^5 (d+e x)+425425 b^6 d^4 (d+e x)^2-464100 b^6 d^3 (d+e x)^3+294525 b^6 d^2 (d+e x)^4+15015 b^6 (d+e x)^6-102102 b^6 d (d+e x)^5\right )}{255255 e^6 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(51051*b^6*d^6 - 306306*a*b^5*d^5*e + 765765*a^2*b^4*d^4*e^2 - 10
21020*a^3*b^3*d^3*e^3 + 765765*a^4*b^2*d^2*e^4 - 306306*a^5*b*d*e^5 + 51051*a^6*e^6 - 218790*b^6*d^5*(d + e*x)
+ 1093950*a*b^5*d^4*e*(d + e*x) - 2187900*a^2*b^4*d^3*e^2*(d + e*x) + 2187900*a^3*b^3*d^2*e^3*(d + e*x) - 109
3950*a^4*b^2*d*e^4*(d + e*x) + 218790*a^5*b*e^5*(d + e*x) + 425425*b^6*d^4*(d + e*x)^2 - 1701700*a*b^5*d^3*e*(
d + e*x)^2 + 2552550*a^2*b^4*d^2*e^2*(d + e*x)^2 - 1701700*a^3*b^3*d*e^3*(d + e*x)^2 + 425425*a^4*b^2*e^4*(d +
e*x)^2 - 464100*b^6*d^3*(d + e*x)^3 + 1392300*a*b^5*d^2*e*(d + e*x)^3 - 1392300*a^2*b^4*d*e^2*(d + e*x)^3 + 4
64100*a^3*b^3*e^3*(d + e*x)^3 + 294525*b^6*d^2*(d + e*x)^4 - 589050*a*b^5*d*e*(d + e*x)^4 + 294525*a^2*b^4*e^2
*(d + e*x)^4 - 102102*b^6*d*(d + e*x)^5 + 102102*a*b^5*e*(d + e*x)^5 + 15015*b^6*(d + e*x)^6))/(255255*e^6*(a*
e + b*e*x))

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fricas [A]  time = 0.44, size = 541, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (15015 \, b^{6} e^{8} x^{8} + 1024 \, b^{6} d^{8} - 8704 \, a b^{5} d^{7} e + 32640 \, a^{2} b^{4} d^{6} e^{2} - 70720 \, a^{3} b^{3} d^{5} e^{3} + 97240 \, a^{4} b^{2} d^{4} e^{4} - 87516 \, a^{5} b d^{3} e^{5} + 51051 \, a^{6} d^{2} e^{6} + 6006 \, {\left (3 \, b^{6} d e^{7} + 17 \, a b^{5} e^{8}\right )} x^{7} + 231 \, {\left (b^{6} d^{2} e^{6} + 544 \, a b^{5} d e^{7} + 1275 \, a^{2} b^{4} e^{8}\right )} x^{6} - 42 \, {\left (6 \, b^{6} d^{3} e^{5} - 51 \, a b^{5} d^{2} e^{6} - 8925 \, a^{2} b^{4} d e^{7} - 11050 \, a^{3} b^{3} e^{8}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{4} e^{4} - 68 \, a b^{5} d^{3} e^{5} + 255 \, a^{2} b^{4} d^{2} e^{6} + 17680 \, a^{3} b^{3} d e^{7} + 12155 \, a^{4} b^{2} e^{8}\right )} x^{4} - 10 \, {\left (32 \, b^{6} d^{5} e^{3} - 272 \, a b^{5} d^{4} e^{4} + 1020 \, a^{2} b^{4} d^{3} e^{5} - 2210 \, a^{3} b^{3} d^{2} e^{6} - 60775 \, a^{4} b^{2} d e^{7} - 21879 \, a^{5} b e^{8}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{6} e^{2} - 1088 \, a b^{5} d^{5} e^{3} + 4080 \, a^{2} b^{4} d^{4} e^{4} - 8840 \, a^{3} b^{3} d^{3} e^{5} + 12155 \, a^{4} b^{2} d^{2} e^{6} + 116688 \, a^{5} b d e^{7} + 17017 \, a^{6} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{7} e - 2176 \, a b^{5} d^{6} e^{2} + 8160 \, a^{2} b^{4} d^{5} e^{3} - 17680 \, a^{3} b^{3} d^{4} e^{4} + 24310 \, a^{4} b^{2} d^{3} e^{5} - 21879 \, a^{5} b d^{2} e^{6} - 51051 \, a^{6} 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((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/255255*(15015*b^6*e^8*x^8 + 1024*b^6*d^8 - 8704*a*b^5*d^7*e + 32640*a^2*b^4*d^6*e^2 - 70720*a^3*b^3*d^5*e^3
+ 97240*a^4*b^2*d^4*e^4 - 87516*a^5*b*d^3*e^5 + 51051*a^6*d^2*e^6 + 6006*(3*b^6*d*e^7 + 17*a*b^5*e^8)*x^7 + 23
1*(b^6*d^2*e^6 + 544*a*b^5*d*e^7 + 1275*a^2*b^4*e^8)*x^6 - 42*(6*b^6*d^3*e^5 - 51*a*b^5*d^2*e^6 - 8925*a^2*b^4
*d*e^7 - 11050*a^3*b^3*e^8)*x^5 + 35*(8*b^6*d^4*e^4 - 68*a*b^5*d^3*e^5 + 255*a^2*b^4*d^2*e^6 + 17680*a^3*b^3*d
*e^7 + 12155*a^4*b^2*e^8)*x^4 - 10*(32*b^6*d^5*e^3 - 272*a*b^5*d^4*e^4 + 1020*a^2*b^4*d^3*e^5 - 2210*a^3*b^3*d
^2*e^6 - 60775*a^4*b^2*d*e^7 - 21879*a^5*b*e^8)*x^3 + 3*(128*b^6*d^6*e^2 - 1088*a*b^5*d^5*e^3 + 4080*a^2*b^4*d
^4*e^4 - 8840*a^3*b^3*d^3*e^5 + 12155*a^4*b^2*d^2*e^6 + 116688*a^5*b*d*e^7 + 17017*a^6*e^8)*x^2 - 2*(256*b^6*d
^7*e - 2176*a*b^5*d^6*e^2 + 8160*a^2*b^4*d^5*e^3 - 17680*a^3*b^3*d^4*e^4 + 24310*a^4*b^2*d^3*e^5 - 21879*a^5*b
*d^2*e^6 - 51051*a^6*d*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.38, size = 1609, normalized size = 4.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(1531530*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d^2*e^(-1)*sgn(b*x + a) + 765765*(3*(x*e + d)^(5
/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d^2*e^(-2)*sgn(b*x + a) + 437580*(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^3*d^2*e^(-3)*sgn(b*x + a) + 364
65*(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^4*d^2*e^(-4)*sgn(b*x + a) + 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*b^5*d^2*e^(-5)*
sgn(b*x + a) + 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)*b^6*d^2*e^(-6)*sgn(
b*x + a) + 612612*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b*d*e^(-1)*sgn(b*x + a
) + 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^4*b^2*
d*e^(-2)*sgn(b*x + a) + 97240*(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^3*b^3*d*e^(-3)*sgn(b*x + a) + 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^4*d*e^(-4)*sgn(b*x + a) + 3060*(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^5*d*e^(-5)*sgn(b*x + a) + 238*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(1
1/2)*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^6*d*e^(-6)*sgn(b*x + a) + 765765*sqrt(x*e + d)*a^6*d^2*sgn(b*x + a) + 5
10510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*d*sgn(b*x + a) + 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^5*b*e^(-1)*sgn(b*x + a) + 36465*(35*(x*e + d)^(9/2) - 1
80*(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^2*e^(-
2)*sgn(b*x + a) + 22100*(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^3*e^(-3)*sgn(b*x + a) + 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^2*b^4*e^(-4)*sgn(b*x + a) + 714*(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^5*e^(-5)*sgn(b*x
+ a) + 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^6*e^(-6)*sgn(b*x + a) + 51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^
(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.05, size = 393, normalized size = 1.05 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 b^{6} e^{6} x^{6}+102102 a \,b^{5} e^{6} x^{5}-12012 b^{6} d \,e^{5} x^{5}+294525 a^{2} b^{4} e^{6} x^{4}-78540 a \,b^{5} d \,e^{5} x^{4}+9240 b^{6} d^{2} e^{4} x^{4}+464100 a^{3} b^{3} e^{6} x^{3}-214200 a^{2} b^{4} d \,e^{5} x^{3}+57120 a \,b^{5} d^{2} e^{4} x^{3}-6720 b^{6} d^{3} e^{3} x^{3}+425425 a^{4} b^{2} e^{6} x^{2}-309400 a^{3} b^{3} d \,e^{5} x^{2}+142800 a^{2} b^{4} d^{2} e^{4} x^{2}-38080 a \,b^{5} d^{3} e^{3} x^{2}+4480 b^{6} d^{4} e^{2} x^{2}+218790 a^{5} b \,e^{6} x -243100 a^{4} b^{2} d \,e^{5} x +176800 a^{3} b^{3} d^{2} e^{4} x -81600 a^{2} b^{4} d^{3} e^{3} x +21760 a \,b^{5} d^{4} e^{2} x -2560 b^{6} d^{5} e x +51051 a^{6} e^{6}-87516 a^{5} b d \,e^{5}+97240 a^{4} b^{2} d^{2} e^{4}-70720 a^{3} b^{3} d^{3} e^{3}+32640 a^{2} b^{4} d^{4} e^{2}-8704 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{255255 \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(e*x+d)^(5/2)*(15015*b^6*e^6*x^6+102102*a*b^5*e^6*x^5-12012*b^6*d*e^5*x^5+294525*a^2*b^4*e^6*x^4-7854
0*a*b^5*d*e^5*x^4+9240*b^6*d^2*e^4*x^4+464100*a^3*b^3*e^6*x^3-214200*a^2*b^4*d*e^5*x^3+57120*a*b^5*d^2*e^4*x^3
-6720*b^6*d^3*e^3*x^3+425425*a^4*b^2*e^6*x^2-309400*a^3*b^3*d*e^5*x^2+142800*a^2*b^4*d^2*e^4*x^2-38080*a*b^5*d
^3*e^3*x^2+4480*b^6*d^4*e^2*x^2+218790*a^5*b*e^6*x-243100*a^4*b^2*d*e^5*x+176800*a^3*b^3*d^2*e^4*x-81600*a^2*b
^4*d^3*e^3*x+21760*a*b^5*d^4*e^2*x-2560*b^6*d^5*e*x+51051*a^6*e^6-87516*a^5*b*d*e^5+97240*a^4*b^2*d^2*e^4-7072
0*a^3*b^3*d^3*e^3+32640*a^2*b^4*d^4*e^2-8704*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.82, size = 921, normalized size = 2.45 \begin {gather*} \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} a}{45045 \, e^{6}} + \frac {2 \, {\left (45045 \, b^{5} e^{8} x^{8} + 3072 \, b^{5} d^{8} - 21760 \, a b^{4} d^{7} e + 65280 \, a^{2} b^{3} d^{6} e^{2} - 106080 \, a^{3} b^{2} d^{5} e^{3} + 97240 \, a^{4} b d^{4} e^{4} - 43758 \, a^{5} d^{3} e^{5} + 3003 \, {\left (18 \, b^{5} d e^{7} + 85 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (3 \, b^{5} d^{2} e^{6} + 1360 \, a b^{4} d e^{7} + 2550 \, a^{2} b^{3} e^{8}\right )} x^{6} - 63 \, {\left (12 \, b^{5} d^{3} e^{5} - 85 \, a b^{4} d^{2} e^{6} - 11900 \, a^{2} b^{3} d e^{7} - 11050 \, a^{3} b^{2} e^{8}\right )} x^{5} + 35 \, {\left (24 \, b^{5} d^{4} e^{4} - 170 \, a b^{4} d^{3} e^{5} + 510 \, a^{2} b^{3} d^{2} e^{6} + 26520 \, a^{3} b^{2} d e^{7} + 12155 \, a^{4} b e^{8}\right )} x^{4} - 5 \, {\left (192 \, b^{5} d^{5} e^{3} - 1360 \, a b^{4} d^{4} e^{4} + 4080 \, a^{2} b^{3} d^{3} e^{5} - 6630 \, a^{3} b^{2} d^{2} e^{6} - 121550 \, a^{4} b d e^{7} - 21879 \, a^{5} e^{8}\right )} x^{3} + 3 \, {\left (384 \, b^{5} d^{6} e^{2} - 2720 \, a b^{4} d^{5} e^{3} + 8160 \, a^{2} b^{3} d^{4} e^{4} - 13260 \, a^{3} b^{2} d^{3} e^{5} + 12155 \, a^{4} b d^{2} e^{6} + 58344 \, a^{5} d e^{7}\right )} x^{2} - {\left (1536 \, b^{5} d^{7} e - 10880 \, a b^{4} d^{6} e^{2} + 32640 \, a^{2} b^{3} d^{5} e^{3} - 53040 \, a^{3} b^{2} d^{4} e^{4} + 48620 \, a^{4} b d^{3} e^{5} - 21879 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d} b}{765765 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)*a/e^6 + 2/765765*(45045*b^5*e^8*x^8 + 3072*b^5*d^8 - 21760*a*b^4*d^7*
e + 65280*a^2*b^3*d^6*e^2 - 106080*a^3*b^2*d^5*e^3 + 97240*a^4*b*d^4*e^4 - 43758*a^5*d^3*e^5 + 3003*(18*b^5*d*
e^7 + 85*a*b^4*e^8)*x^7 + 231*(3*b^5*d^2*e^6 + 1360*a*b^4*d*e^7 + 2550*a^2*b^3*e^8)*x^6 - 63*(12*b^5*d^3*e^5 -
85*a*b^4*d^2*e^6 - 11900*a^2*b^3*d*e^7 - 11050*a^3*b^2*e^8)*x^5 + 35*(24*b^5*d^4*e^4 - 170*a*b^4*d^3*e^5 + 51
0*a^2*b^3*d^2*e^6 + 26520*a^3*b^2*d*e^7 + 12155*a^4*b*e^8)*x^4 - 5*(192*b^5*d^5*e^3 - 1360*a*b^4*d^4*e^4 + 408
0*a^2*b^3*d^3*e^5 - 6630*a^3*b^2*d^2*e^6 - 121550*a^4*b*d*e^7 - 21879*a^5*e^8)*x^3 + 3*(384*b^5*d^6*e^2 - 2720
*a*b^4*d^5*e^3 + 8160*a^2*b^3*d^4*e^4 - 13260*a^3*b^2*d^3*e^5 + 12155*a^4*b*d^2*e^6 + 58344*a^5*d*e^7)*x^2 - (
1536*b^5*d^7*e - 10880*a*b^4*d^6*e^2 + 32640*a^2*b^3*d^5*e^3 - 53040*a^3*b^2*d^4*e^4 + 48620*a^4*b*d^3*e^5 - 2
1879*a^5*d^2*e^6)*x)*sqrt(e*x + d)*b/e^7

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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