3.2108 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=262 \[ -\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^{7/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^{9/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt{d+e x}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^{3/2}} \]

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)*(d + e*x)^(9/2
)) + (8*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x
)^(7/2)) - (12*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)
*(d + e*x)^(5/2)) + (8*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a
+ b*x)*(d + e*x)^(3/2)) - (2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*S
qrt[d + e*x])

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Rubi [A]  time = 0.305382, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^{7/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^{9/2}}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt{d+e x}}+\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(11/2),x]

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)*(d + e*x)^(9/2
)) + (8*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x
)^(7/2)) - (12*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)
*(d + e*x)^(5/2)) + (8*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a
+ b*x)*(d + e*x)^(3/2)) - (2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*S
qrt[d + e*x])

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Rubi in Sympy [A]  time = 35.3948, size = 212, normalized size = 0.81 \[ - \frac{128 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 e^{4} \left (d + e x\right )^{\frac{3}{2}}} + \frac{256 b^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 e^{5} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} - \frac{32 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 e^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{16 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e \left (d + e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(11/2),x)

[Out]

-128*b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(105*e**4*(d + e*x)**(3/2)) + 256*b**
3*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*e**5*(a + b*x)*(d + e*x)**(3
/2)) - 32*b**2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*e**3*(d + e*x
)**(5/2)) - 16*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*e**2*(d + e*x)**(7/2))
- 2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9*e*(d + e*x)**(9/2))

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Mathematica [A]  time = 0.257326, size = 119, normalized size = 0.45 \[ \frac{2 \sqrt{(a+b x)^2} \left (420 b^3 (d+e x)^3 (b d-a e)-378 b^2 (d+e x)^2 (b d-a e)^2+180 b (d+e x) (b d-a e)^3-35 (b d-a e)^4-315 b^4 (d+e x)^4\right )}{315 e^5 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-35*(b*d - a*e)^4 + 180*b*(b*d - a*e)^3*(d + e*x) - 378*b^
2*(b*d - a*e)^2*(d + e*x)^2 + 420*b^3*(b*d - a*e)*(d + e*x)^3 - 315*b^4*(d + e*x
)^4))/(315*e^5*(a + b*x)*(d + e*x)^(9/2))

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Maple [A]  time = 0.012, size = 202, normalized size = 0.8 \[ -{\frac{630\,{x}^{4}{b}^{4}{e}^{4}+840\,{x}^{3}a{b}^{3}{e}^{4}+1680\,{x}^{3}{b}^{4}d{e}^{3}+756\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+1008\,{x}^{2}a{b}^{3}d{e}^{3}+2016\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+360\,x{a}^{3}b{e}^{4}+432\,x{a}^{2}{b}^{2}d{e}^{3}+576\,xa{b}^{3}{d}^{2}{e}^{2}+1152\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}+80\,{a}^{3}bd{e}^{3}+96\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+128\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{315\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(11/2),x)

[Out]

-2/315/(e*x+d)^(9/2)*(315*b^4*e^4*x^4+420*a*b^3*e^4*x^3+840*b^4*d*e^3*x^3+378*a^
2*b^2*e^4*x^2+504*a*b^3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+180*a^3*b*e^4*x+216*a^2*b
^2*d*e^3*x+288*a*b^3*d^2*e^2*x+576*b^4*d^3*e*x+35*a^4*e^4+40*a^3*b*d*e^3+48*a^2*
b^2*d^2*e^2+64*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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Maxima [A]  time = 0.739814, size = 501, normalized size = 1.91 \[ -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} a}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} - \frac{2 \,{\left (315 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} + 48 \, a b^{2} d^{3} e + 24 \, a^{2} b d^{2} e^{2} + 10 \, a^{3} d e^{3} + 105 \,{\left (8 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 63 \,{\left (16 \, b^{3} d^{2} e^{2} + 6 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} + 9 \,{\left (64 \, b^{3} d^{3} e + 24 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + 5 \, a^{3} e^{4}\right )} x\right )} b}{315 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

-2/315*(105*b^3*e^3*x^3 + 16*b^3*d^3 + 24*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 35*a^3*
e^3 + 63*(2*b^3*d*e^2 + 3*a*b^2*e^3)*x^2 + 9*(8*b^3*d^2*e + 12*a*b^2*d*e^2 + 15*
a^2*b*e^3)*x)*a/((e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)
*sqrt(e*x + d)) - 2/315*(315*b^3*e^4*x^4 + 128*b^3*d^4 + 48*a*b^2*d^3*e + 24*a^2
*b*d^2*e^2 + 10*a^3*d*e^3 + 105*(8*b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 63*(16*b^3*d^2
*e^2 + 6*a*b^2*d*e^3 + 3*a^2*b*e^4)*x^2 + 9*(64*b^3*d^3*e + 24*a*b^2*d^2*e^2 + 1
2*a^2*b*d*e^3 + 5*a^3*e^4)*x)*b/((e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*
e^6*x + d^4*e^5)*sqrt(e*x + d))

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Fricas [A]  time = 0.279777, size = 302, normalized size = 1.15 \[ -\frac{2 \,{\left (315 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 64 \, a b^{3} d^{3} e + 48 \, a^{2} b^{2} d^{2} e^{2} + 40 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 420 \,{\left (2 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 126 \,{\left (8 \, b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 36 \,{\left (16 \, b^{4} d^{3} e + 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x\right )}}{315 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

-2/315*(315*b^4*e^4*x^4 + 128*b^4*d^4 + 64*a*b^3*d^3*e + 48*a^2*b^2*d^2*e^2 + 40
*a^3*b*d*e^3 + 35*a^4*e^4 + 420*(2*b^4*d*e^3 + a*b^3*e^4)*x^3 + 126*(8*b^4*d^2*e
^2 + 4*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 + 36*(16*b^4*d^3*e + 8*a*b^3*d^2*e^2 + 6
*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)/((e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3
*e^6*x + d^4*e^5)*sqrt(e*x + d))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335955, size = 414, normalized size = 1.58 \[ -\frac{2 \,{\left (315 \,{\left (x e + d\right )}^{4} b^{4}{\rm sign}\left (b x + a\right ) - 420 \,{\left (x e + d\right )}^{3} b^{4} d{\rm sign}\left (b x + a\right ) + 378 \,{\left (x e + d\right )}^{2} b^{4} d^{2}{\rm sign}\left (b x + a\right ) - 180 \,{\left (x e + d\right )} b^{4} d^{3}{\rm sign}\left (b x + a\right ) + 35 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 420 \,{\left (x e + d\right )}^{3} a b^{3} e{\rm sign}\left (b x + a\right ) - 756 \,{\left (x e + d\right )}^{2} a b^{3} d e{\rm sign}\left (b x + a\right ) + 540 \,{\left (x e + d\right )} a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 140 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 378 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right ) - 540 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 210 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 180 \,{\left (x e + d\right )} a^{3} b e^{3}{\rm sign}\left (b x + a\right ) - 140 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 35 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{315 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

-2/315*(315*(x*e + d)^4*b^4*sign(b*x + a) - 420*(x*e + d)^3*b^4*d*sign(b*x + a)
+ 378*(x*e + d)^2*b^4*d^2*sign(b*x + a) - 180*(x*e + d)*b^4*d^3*sign(b*x + a) +
35*b^4*d^4*sign(b*x + a) + 420*(x*e + d)^3*a*b^3*e*sign(b*x + a) - 756*(x*e + d)
^2*a*b^3*d*e*sign(b*x + a) + 540*(x*e + d)*a*b^3*d^2*e*sign(b*x + a) - 140*a*b^3
*d^3*e*sign(b*x + a) + 378*(x*e + d)^2*a^2*b^2*e^2*sign(b*x + a) - 540*(x*e + d)
*a^2*b^2*d*e^2*sign(b*x + a) + 210*a^2*b^2*d^2*e^2*sign(b*x + a) + 180*(x*e + d)
*a^3*b*e^3*sign(b*x + a) - 140*a^3*b*d*e^3*sign(b*x + a) + 35*a^4*e^4*sign(b*x +
a))*e^(-5)/(x*e + d)^(9/2)