3.2131 \(\int \frac{(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{(d+e x)^{5/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(5*e*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(5*e*(a + b*x)*(d + e*x)^(3/2))/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x
)^(5/2)/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e*(b*d - a*e)^(3/2)*(a + b*x)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])

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Rubi [A]  time = 0.351281, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^{5/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*e*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(5*e*(a + b*x)*(d + e*x)^(3/2))/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x
)^(5/2)/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e*(b*d - a*e)^(3/2)*(a + b*x)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.310755, size = 121, normalized size = 0.61 \[ \frac{(a+b x) \left (\frac{\sqrt{d+e x} \left (-\frac{3 (b d-a e)^2}{a+b x}+2 e (7 b d-6 a e)+2 b e^2 x\right )}{3 b^3}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 409, normalized size = 2.1 \[{\frac{ \left ( bx+a \right ) ^{2}}{3\,{b}^{3}} \left ( 2\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}x{b}^{2}e+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}-30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{2}d{e}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{b}^{3}{d}^{2}e+2\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe-12\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xab{e}^{2}+12\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{2}de+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}-30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{2}{d}^{2}e-15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}+18\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

1/3*(2*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*b^2*e+15*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*x*a^2*b*e^3-30*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b
^2*d*e^2+15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*b^3*d^2*e+2*(b*(a*e-b*
d))^(1/2)*(e*x+d)^(3/2)*a*b*e-12*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b*e^2+12*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*b^2*d*e+15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*
d))^(1/2))*a^3*e^3-30*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b*d*e^2+15
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*b^2*d^2*e-15*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(1/2)*a^2*e^2+18*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b*d*e-3*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(1/2)*b^2*d^2)*(b*x+a)^2/(b*(a*e-b*d))^(1/2)/b^3/((b*x+a)^2)^(3
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.334187, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[-1/6*(15*(a*b*d*e - a^2*e^2 + (b^2*d*e - a*b*e^2)*x)*sqrt((b*d - a*e)/b)*log((b
*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(2*b^
2*e^2*x^2 - 3*b^2*d^2 + 20*a*b*d*e - 15*a^2*e^2 + 2*(7*b^2*d*e - 5*a*b*e^2)*x)*s
qrt(e*x + d))/(b^4*x + a*b^3), -1/3*(15*(a*b*d*e - a^2*e^2 + (b^2*d*e - a*b*e^2)
*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (2*b^2*e^2
*x^2 - 3*b^2*d^2 + 20*a*b*d*e - 15*a^2*e^2 + 2*(7*b^2*d*e - 5*a*b*e^2)*x)*sqrt(e
*x + d))/(b^4*x + a*b^3)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.319987, size = 366, normalized size = 1.85 \[ -\frac{5 \,{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt{-b^{2} d + a b e} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (\sqrt{x e + d} b^{2} d^{2} e^{2} - 2 \, \sqrt{x e + d} a b d e^{3} + \sqrt{x e + d} a^{2} e^{4}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{4} + 6 \, \sqrt{x e + d} b^{4} d e^{4} - 6 \, \sqrt{x e + d} a b^{3} e^{5}\right )} e^{\left (-3\right )}}{3 \, b^{6}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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)^(3/2),x, algorithm="giac")

[Out]

-5*(b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*
b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + (
sqrt(x*e + d)*b^2*d^2*e^2 - 2*sqrt(x*e + d)*a*b*d*e^3 + sqrt(x*e + d)*a^2*e^4)*e
^(-1)/(((x*e + d)*b - b*d + a*e)*b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/3
*((x*e + d)^(3/2)*b^4*e^4 + 6*sqrt(x*e + d)*b^4*d*e^4 - 6*sqrt(x*e + d)*a*b^3*e^
5)*e^(-3)/(b^6*sign(-(x*e + d)*b*e + b*d*e - a*e^2))