3.2142 \(\int \frac{a+b x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=225 \[ \frac{5 e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{5 e \sqrt{d+e x}}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(-5*e^2*Sqrt[d + e*x])/(8*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d
+ e*x]/(3*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*Sqrt[d +
e*x])/(12*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^3*(a +
b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(7
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.418593, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{5 e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{5 e \sqrt{d+e x}}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-5*e^2*Sqrt[d + e*x])/(8*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d
+ e*x]/(3*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*Sqrt[d +
e*x])/(12*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^3*(a +
b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(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)/(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 = 0.428221, size = 146, normalized size = 0.65 \[ \frac{(a+b x) \left (\frac{5 e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (33 a^2 e^2+2 a b e (20 e x-13 d)+b^2 \left (8 d^2-10 d e x+15 e^2 x^2\right )\right )}{3 (a+b x)^3 (b d-a e)^3}\right )}{8 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*(33*a^2*e^2 + 2*a*b*e*(-13*d + 20*e*x) + b^2*(8*d^2
- 10*d*e*x + 15*e^2*x^2)))/(3*(b*d - a*e)^3*(a + b*x)^3) + (5*e^3*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(7/2))))/(8*Sqrt[(a + b
*x)^2])

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Maple [B]  time = 0.017, size = 334, normalized size = 1.5 \[{\frac{ \left ( bx+a \right ) ^{2}}{24\, \left ( ae-bd \right ) ^{3}} \left ( 15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}{b}^{3}{e}^{3}+45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}a{b}^{2}{e}^{3}+15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}{b}^{2}{e}^{2}+45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}+40\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xab{e}^{2}-10\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{2}de+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}+33\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}-26\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde+8\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*b^3*e^3+45*arctan((e*x+
d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^2*e^3+15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*x^2*b^2*e^2+45*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b*e^3+40*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b*e^2-10*(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+33*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(1/2)*a^2*e^2-26*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b*d*e+8*(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)^3/
((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)/((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 = 0.308826, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 26 \, a b d e + 33 \, a^{2} e^{2} - 10 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 26 \, a b d e + 33 \, a^{2} e^{2} - 10 \,{\left (b^{2} d e - 4 \, a b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{24 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(15*b^2*e^2*x^2 + 8*b^2*d^2 - 26*a*b*d*e + 33*a^2*e^2 - 10*(b^2*d*e -
4*a*b*e^2)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(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^3 - 3*a^4*b^2*d^2*e + 3
*a^5*b*d*e^2 - a^6*e^3 + (b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^
3)*x^3 + 3*(a*b^5*d^3 - 3*a^2*b^4*d^2*e + 3*a^3*b^3*d*e^2 - a^4*b^2*e^3)*x^2 + 3
*(a^2*b^4*d^3 - 3*a^3*b^3*d^2*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3)*x)*sqrt(b^2*d - a
*b*e)), -1/24*((15*b^2*e^2*x^2 + 8*b^2*d^2 - 26*a*b*d*e + 33*a^2*e^2 - 10*(b^2*d
*e - 4*a*b*e^2)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 15*(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)*s
qrt(e*x + d))))/((a^3*b^3*d^3 - 3*a^4*b^2*d^2*e + 3*a^5*b*d*e^2 - a^6*e^3 + (b^6
*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*x^3 + 3*(a*b^5*d^3 - 3*a^2
*b^4*d^2*e + 3*a^3*b^3*d*e^2 - a^4*b^2*e^3)*x^2 + 3*(a^2*b^4*d^3 - 3*a^3*b^3*d^2
*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3)*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)/(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 = 0.308239, size = 563, normalized size = 2.5 \[ \frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 33 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 66 \, \sqrt{x e + d} a b d e^{4} + 33 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d^3*sign(-(x*e + d)*b
*e + b*d*e - a*e^2) - 3*a*b^2*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3*a^2
*b*d*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*e^3*sign(-(x*e + d)*b*e + b*
d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) + 1/24*(15*(x*e + d)^(5/2)*b^2*e^3 - 40*(x*e
+ d)^(3/2)*b^2*d*e^3 + 33*sqrt(x*e + d)*b^2*d^2*e^3 + 40*(x*e + d)^(3/2)*a*b*e^
4 - 66*sqrt(x*e + d)*a*b*d*e^4 + 33*sqrt(x*e + d)*a^2*e^5)/((b^3*d^3*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) - 3*a*b^2*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
3*a^2*b*d*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*e^3*sign(-(x*e + d)*b*
e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^3)