3.1716 \(\int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=278 \[ -\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d
+ e*x]/(4*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*Sqrt[d
+ e*x])/(24*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*S
qrt[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e
^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d
- a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.503308, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d
+ e*x]/(4*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*Sqrt[d
+ e*x])/(24*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*S
qrt[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e
^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d
- a*e)^(9/2)*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(1/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.68359, size = 166, normalized size = 0.6 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (70 e^2 (a+b x)^2 (a e-b d)+56 e (a+b x) (b d-a e)^2-48 (b d-a e)^3+105 e^3 (a+b x)^3\right )}{3 (a+b x)^4 (b d-a e)^4}-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{9/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((a + b*x)^5*((Sqrt[d + e*x]*(-48*(b*d - a*e)^3 + 56*e*(b*d - a*e)^2*(a + b*x) +
70*e^2*(-(b*d) + a*e)*(a + b*x)^2 + 105*e^3*(a + b*x)^3))/(3*(b*d - a*e)^4*(a +
b*x)^4) - (35*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b
*d - a*e)^(9/2))))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.017, size = 497, normalized size = 1.8 \[{\frac{bx+a}{192\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{4}{b}^{4}{e}^{4}+420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{3}{b}^{3}{e}^{3}+630\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+385\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{3}-70\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}{b}^{3}d{e}^{2}+420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+511\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{a}^{2}b{e}^{3}-252\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xa{b}^{2}d{e}^{2}+56\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{3}{d}^{2}e+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}+279\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}-326\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}+200\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e-48\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

1/192*(105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^4*e^4+420*arctan((e
*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^3*e^4+105*(b*(a*e-b*d))^(1/2)*(e*x+d)
^(1/2)*x^3*b^3*e^3+630*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*e
^4+385*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a*b^2*e^3-70*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*x^2*b^3*d*e^2+420*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*
b*e^4+511*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^2*b*e^3-252*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(1/2)*x*a*b^2*d*e^2+56*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*b^3*d^2*e+105
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*e^4+279*(b*(a*e-b*d))^(1/2)*(e*
x+d)^(1/2)*a^3*e^3-326*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2+200*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e-48*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*
d^3)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)^4/((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(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22855, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/384*(2*(105*b^3*e^3*x^3 - 48*b^3*d^3 + 200*a*b^2*d^2*e - 326*a^2*b*d*e^2 + 27
9*a^3*e^3 - 35*(2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 + 7*(8*b^3*d^2*e - 36*a*b^2*d*e^
2 + 73*a^2*b*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 105*(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^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2 - 4*a^7*b*d*e^3 + a^8*e^4 + (b^8*d^
4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4)*x^4 + 4*(
a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3*b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a^5*b^3*e^4)
*x^3 + 6*(a^2*b^6*d^4 - 4*a^3*b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b^3*d*e^3 +
a^6*b^2*e^4)*x^2 + 4*(a^3*b^5*d^4 - 4*a^4*b^4*d^3*e + 6*a^5*b^3*d^2*e^2 - 4*a^6*
b^2*d*e^3 + a^7*b*e^4)*x)*sqrt(b^2*d - a*b*e)), 1/192*((105*b^3*e^3*x^3 - 48*b^3
*d^3 + 200*a*b^2*d^2*e - 326*a^2*b*d*e^2 + 279*a^3*e^3 - 35*(2*b^3*d*e^2 - 11*a*
b^2*e^3)*x^2 + 7*(8*b^3*d^2*e - 36*a*b^2*d*e^2 + 73*a^2*b*e^3)*x)*sqrt(-b^2*d +
a*b*e)*sqrt(e*x + d) - 105*(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^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2 - 4*a^7*b*d*e^3 + a^8*e^
4 + (b^8*d^4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4
)*x^4 + 4*(a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3*b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a
^5*b^3*e^4)*x^3 + 6*(a^2*b^6*d^4 - 4*a^3*b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b
^3*d*e^3 + a^6*b^2*e^4)*x^2 + 4*(a^3*b^5*d^4 - 4*a^4*b^4*d^3*e + 6*a^5*b^3*d^2*e
^2 - 4*a^6*b^2*d*e^3 + a^7*b*e^4)*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(1/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.240812, size = 757, normalized size = 2.72 \[ -\frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{105 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 385 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 1022 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 837 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 837 \, \sqrt{x e + d} a^{2} b d e^{6} + 279 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

-35/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^4*sign(-(x*e + d
)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*
a^2*b^2*d^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) + a^4*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-
b^2*d + a*b*e)) - 1/192*(105*(x*e + d)^(7/2)*b^3*e^4 - 385*(x*e + d)^(5/2)*b^3*d
*e^4 + 511*(x*e + d)^(3/2)*b^3*d^2*e^4 - 279*sqrt(x*e + d)*b^3*d^3*e^4 + 385*(x*
e + d)^(5/2)*a*b^2*e^5 - 1022*(x*e + d)^(3/2)*a*b^2*d*e^5 + 837*sqrt(x*e + d)*a*
b^2*d^2*e^5 + 511*(x*e + d)^(3/2)*a^2*b*e^6 - 837*sqrt(x*e + d)*a^2*b*d*e^6 + 27
9*sqrt(x*e + d)*a^3*e^7)/((b^4*d^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a*b^
3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*a^2*b^2*d^2*e^2*sign(-(x*e + d)
*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a^4
*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^4)