3.2140 \(\int \frac{(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{e^2 \sqrt{d+e x}}{8 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

[Out]

-(d + e*x)^(3/2)/(3*b*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)) - (e^2*Sqrt[d + e*x])/(8*
b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(4*b^2*(a + b
*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[b*d - a*e]])/(8*b^(5/2)*(b*d - a*e)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)

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Rubi [A]  time = 0.36031, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{e^2 \sqrt{d+e x}}{8 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

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]

-(d + e*x)^(3/2)/(3*b*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)) - (e^2*Sqrt[d + e*x])/(8*
b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(4*b^2*(a + b
*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[b*d - a*e]])/(8*b^(5/2)*(b*d - a*e)^(3/2)*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.35388, size = 147, normalized size = 0.71 \[ \frac{(a+b x) \left (\frac{\sqrt{d+e x} \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (-\left (8 d^2+14 d e x+3 e^2 x^2\right )\right )\right )}{3 b^2 (a+b x)^3 (b d-a e)}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} (b d-a e)^{3/2}}\right )}{8 \sqrt{(a+b x)^2}} \]

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]

((a + b*x)*((Sqrt[d + e*x]*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) - b^2*(8*d^2 + 14*d*
e*x + 3*e^2*x^2)))/(3*b^2*(b*d - a*e)*(a + b*x)^3) + (e^3*ArcTanh[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*(b*d - a*e)^(3/2))))/(8*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.024, size = 326, normalized size = 1.6 \[{\frac{ \left ( bx+a \right ) ^{2}}{ \left ( 24\,ae-24\,bd \right ){b}^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}{b}^{3}{e}^{3}+9\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}a{b}^{2}{e}^{3}+3\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{2}+9\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}-8\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe+8\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}d+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}+6\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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]

1/24*(3*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*b^3*e^3+9*arctan((e*x+d)
^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^2*e^3+3*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*
b^2+9*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b*e^3-8*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(3/2)*a*b*e+8*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^2*d+3*arctan((e*x+
d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*e^3-3*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*
e^2+6*(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)/(a*e-b*d)/b^2/((b*x+a)^2)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313076, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 2 \, a b d e - 3 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} d - a^{4} b^{2} e +{\left (b^{6} d - a b^{5} e\right )} x^{3} + 3 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 2 \, a b d e - 3 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (a^{3} b^{3} d - a^{4} b^{2} e +{\left (b^{6} d - a b^{5} e\right )} x^{3} + 3 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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]

[-1/48*(2*(3*b^2*e^2*x^2 + 8*b^2*d^2 - 2*a*b*d*e - 3*a^2*e^2 + 2*(7*b^2*d*e - 4*
a*b*e^2)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 3*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2
+ 3*a^2*b*e^3*x + a^3*e^3)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(
b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^3*b^3*d - a^4*b^2*e + (b^6*d - a*b
^5*e)*x^3 + 3*(a*b^5*d - a^2*b^4*e)*x^2 + 3*(a^2*b^4*d - a^3*b^3*e)*x)*sqrt(b^2*
d - a*b*e)), -1/24*((3*b^2*e^2*x^2 + 8*b^2*d^2 - 2*a*b*d*e - 3*a^2*e^2 + 2*(7*b^
2*d*e - 4*a*b*e^2)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 3*(b^3*e^3*x^3 + 3*a*
b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)
*sqrt(e*x + d))))/((a^3*b^3*d - a^4*b^2*e + (b^6*d - a*b^5*e)*x^3 + 3*(a*b^5*d -
a^2*b^4*e)*x^2 + 3*(a^2*b^4*d - a^3*b^3*e)*x)*sqrt(-b^2*d + a*b*e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304176, size = 382, normalized size = 1.85 \[ \frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} - 3 \, \sqrt{x e + d} b^{2} d^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} + 6 \, \sqrt{x e + d} a b d e^{4} - 3 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

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]

1/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2) - a*b^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-b^2*d +
a*b*e)) + 1/24*(3*(x*e + d)^(5/2)*b^2*e^3 + 8*(x*e + d)^(3/2)*b^2*d*e^3 - 3*sqrt
(x*e + d)*b^2*d^2*e^3 - 8*(x*e + d)^(3/2)*a*b*e^4 + 6*sqrt(x*e + d)*a*b*d*e^4 -
3*sqrt(x*e + d)*a^2*e^5)/((b^3*d*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a*b^2*e*
sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^3)