3.1680 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=208 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}} \]

[Out]

(2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^4*(a + b*x)*(d + e*x)^(9/2)
) - (6*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)
^(7/2)) + (6*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d
+ e*x)^(5/2)) - (2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4*(a + b*x)*(d + e*x)
^(3/2))

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Rubi [A]  time = 0.204643, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^4*(a + b*x)*(d + e*x)^(9/2)
) - (6*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)
^(7/2)) + (6*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d
+ e*x)^(5/2)) - (2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4*(a + b*x)*(d + e*x)
^(3/2))

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Rubi in Sympy [A]  time = 22.3622, size = 168, normalized size = 0.81 \[ - \frac{16 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{3} \left (d + e x\right )^{\frac{5}{2}}} + \frac{32 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 e^{4} \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e \left (d + e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**3*(d + e*x)**(5/2)) + 32*b**2*(
a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*e**4*(a + b*x)*(d + e*x)**(5/2)
) - 4*b*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**2*(d + e*x)**(7/2)
) - 2*(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 = 0.208277, size = 97, normalized size = 0.47 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \left (189 b^2 (d+e x)^2 (b d-a e)-135 b (d+e x) (b d-a e)^2+35 (b d-a e)^3-105 b^3 (d+e x)^3\right )}{315 e^4 (a+b x)^3 (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a + b*x)^2)^(3/2)*(35*(b*d - a*e)^3 - 135*b*(b*d - a*e)^2*(d + e*x) + 189*b
^2*(b*d - a*e)*(d + e*x)^2 - 105*b^3*(d + e*x)^3))/(315*e^4*(a + b*x)^3*(d + e*x
)^(9/2))

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Maple [A]  time = 0.01, size = 132, normalized size = 0.6 \[ -{\frac{210\,{x}^{3}{b}^{3}{e}^{3}+378\,{x}^{2}a{b}^{2}{e}^{3}+252\,{x}^{2}{b}^{3}d{e}^{2}+270\,x{a}^{2}b{e}^{3}+216\,xa{b}^{2}d{e}^{2}+144\,x{b}^{3}{d}^{2}e+70\,{a}^{3}{e}^{3}+60\,{a}^{2}bd{e}^{2}+48\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{315\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^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)*(105*b^3*e^3*x^3+189*a*b^2*e^3*x^2+126*b^3*d*e^2*x^2+135*a^
2*b*e^3*x+108*a*b^2*d*e^2*x+72*b^3*d^2*e*x+35*a^3*e^3+30*a^2*b*d*e^2+24*a*b^2*d^
2*e+16*b^3*d^3)*((b*x+a)^2)^(3/2)/e^4/(b*x+a)^3

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Maxima [A]  time = 0.743124, size = 215, normalized size = 1.03 \[ -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(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)/((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)*s
qrt(e*x + d))

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Fricas [A]  time = 0.20792, size = 215, normalized size = 1.03 \[ -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[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)/((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)*s
qrt(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**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 = 0.220727, size = 262, normalized size = 1.26 \[ -\frac{2 \,{\left (105 \,{\left (x e + d\right )}^{3} b^{3}{\rm sign}\left (b x + a\right ) - 189 \,{\left (x e + d\right )}^{2} b^{3} d{\rm sign}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 35 \, b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 189 \,{\left (x e + d\right )}^{2} a b^{2} e{\rm sign}\left (b x + a\right ) - 270 \,{\left (x e + d\right )} a b^{2} d e{\rm sign}\left (b x + a\right ) + 105 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} a^{2} b e^{2}{\rm sign}\left (b x + a\right ) - 105 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{315 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(105*(x*e + d)^3*b^3*sign(b*x + a) - 189*(x*e + d)^2*b^3*d*sign(b*x + a)
+ 135*(x*e + d)*b^3*d^2*sign(b*x + a) - 35*b^3*d^3*sign(b*x + a) + 189*(x*e + d)
^2*a*b^2*e*sign(b*x + a) - 270*(x*e + d)*a*b^2*d*e*sign(b*x + a) + 105*a*b^2*d^2
*e*sign(b*x + a) + 135*(x*e + d)*a^2*b*e^2*sign(b*x + a) - 105*a^2*b*d*e^2*sign(
b*x + a) + 35*a^3*e^3*sign(b*x + a))*e^(-4)/(x*e + d)^(9/2)