3.1714 \(\int \frac{(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=270 \[ \frac{3 e^3 \sqrt{d+e x}}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{32 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e \sqrt{d+e x}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

[Out]

(3*e^3*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*
Sqrt[d + e*x])/(8*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^2*Sqrt[d +
e*x])/(32*b^2*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^
(3/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e^4*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(5/2)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.461006, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 e^3 \sqrt{d+e x}}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{32 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e \sqrt{d+e x}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*e^3*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*
Sqrt[d + e*x])/(8*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^2*Sqrt[d +
e*x])/(32*b^2*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^
(3/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e^4*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(5/2)*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((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 = 0.503041, size = 167, normalized size = 0.62 \[ \frac{(a+b x)^5 \left (-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} \left (2 e^2 (a+b x)^2 (b d-a e)+24 e (a+b x) (b d-a e)^2+16 (b d-a e)^3-3 e^3 (a+b x)^3\right )}{b^2 (a+b x)^4 (b d-a e)^2}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^5*(-((Sqrt[d + e*x]*(16*(b*d - a*e)^3 + 24*e*(b*d - a*e)^2*(a + b*x)
+ 2*e^2*(b*d - a*e)*(a + b*x)^2 - 3*e^3*(a + b*x)^3))/(b^2*(b*d - a*e)^2*(a + b*
x)^4)) - (3*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*(b*d
- a*e)^(5/2))))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.031, size = 477, normalized size = 1.8 \[{\frac{bx+a}{64\, \left ( ae-bd \right ) ^{2}{b}^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{4}{b}^{4}{e}^{4}+12\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+3\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{7/2}{b}^{3}+18\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+11\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-11\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}d+12\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}-11\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}+22\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de-11\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}+9\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}-9\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(b*x+a)*(3*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^4*e^4+12*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^3*e^4+3*(b*(a*e-b*d))^(1/2)*(e*x+
d)^(7/2)*b^3+18*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*e^4+11*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-11*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^
3*d+12*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b*e^4-11*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(3/2)*a^2*b*e^2+22*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e-11*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+3*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*a^4*e^4-3*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+9*(b*(a*e-b*d))^(1/2
)*(e*x+d)^(1/2)*a^2*b*d*e^2-9*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e+3*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)/(b*(a*e-b*d))^(1/2)/b^2/(a*e-b*d)^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((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 = 0.225889, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/128*(2*(3*b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 2*a^2*b*d*e^2 - 3*a^3*e
^3 - (2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 - (24*b^3*d^2*e - 44*a*b^2*d*e^2 + 11*a^2*
b*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 +
6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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^4*b^4*d^2 - 2*a^
5*b^3*d*e + a^6*b^2*e^2 + (b^8*d^2 - 2*a*b^7*d*e + a^2*b^6*e^2)*x^4 + 4*(a*b^7*d
^2 - 2*a^2*b^6*d*e + a^3*b^5*e^2)*x^3 + 6*(a^2*b^6*d^2 - 2*a^3*b^5*d*e + a^4*b^4
*e^2)*x^2 + 4*(a^3*b^5*d^2 - 2*a^4*b^4*d*e + a^5*b^3*e^2)*x)*sqrt(b^2*d - a*b*e)
), 1/64*((3*b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 2*a^2*b*d*e^2 - 3*a^3*e^
3 - (2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 - (24*b^3*d^2*e - 44*a*b^2*d*e^2 + 11*a^2*b
*e^3)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 +
6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*arctan(-(b*d - a*e)/(sqrt(-b^2*d +
a*b*e)*sqrt(e*x + d))))/((a^4*b^4*d^2 - 2*a^5*b^3*d*e + a^6*b^2*e^2 + (b^8*d^2
- 2*a*b^7*d*e + a^2*b^6*e^2)*x^4 + 4*(a*b^7*d^2 - 2*a^2*b^6*d*e + a^3*b^5*e^2)*x
^3 + 6*(a^2*b^6*d^2 - 2*a^3*b^5*d*e + a^4*b^4*e^2)*x^2 + 4*(a^3*b^5*d^2 - 2*a^4*
b^4*d*e + a^5*b^3*e^2)*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((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 = 0.249229, size = 576, normalized size = 2.13 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b^{3} d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} b^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 11 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} + 3 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 11 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} + 22 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} - 9 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} + 9 \, \sqrt{x e + d} a^{2} b d e^{6} - 3 \, \sqrt{x e + d} a^{3} e^{7}}{64 \,{\left (b^{4} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b^{3} d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} b^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^2*sign(-(x*e + d)
*b*e + b*d*e - a*e^2) - 2*a*b^3*d*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a^2*b
^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) - 1/64*(3*(x*
e + d)^(7/2)*b^3*e^4 - 11*(x*e + d)^(5/2)*b^3*d*e^4 - 11*(x*e + d)^(3/2)*b^3*d^2
*e^4 + 3*sqrt(x*e + d)*b^3*d^3*e^4 + 11*(x*e + d)^(5/2)*a*b^2*e^5 + 22*(x*e + d)
^(3/2)*a*b^2*d*e^5 - 9*sqrt(x*e + d)*a*b^2*d^2*e^5 - 11*(x*e + d)^(3/2)*a^2*b*e^
6 + 9*sqrt(x*e + d)*a^2*b*d*e^6 - 3*sqrt(x*e + d)*a^3*e^7)/((b^4*d^2*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) - 2*a*b^3*d*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a
^2*b^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^4)