3.1699 \(\int \frac{1}{(d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=219 \[ \frac{2 b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 b (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac{2 (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.292409, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 b (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac{2 (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*(a + b*x))/(5*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2
*b*(a + b*x))/(3*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(2*b^2*(a + b*x))/((b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(2*b^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/((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(1/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.470931, size = 133, normalized size = 0.61 \[ \frac{(a+b x) \left (\frac{6 a^2 e^2-2 a b e (11 d+5 e x)+b^2 \left (46 d^2+70 d e x+30 e^2 x^2\right )}{15 (d+e x)^{5/2} (b d-a e)^3}-\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

((a + b*x)*((6*a^2*e^2 - 2*a*b*e*(11*d + 5*e*x) + b^2*(46*d^2 + 70*d*e*x + 30*e^
2*x^2))/(15*(b*d - a*e)^3*(d + e*x)^(5/2)) - (2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)))/Sqrt[(a + b*x)^2]

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Maple [A]  time = 0.017, size = 202, normalized size = 0.9 \[ -{\frac{2\,bx+2\,a}{15\, \left ( ae-bd \right ) ^{3}} \left ( 15\,{b}^{3}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}+15\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{2}-5\,\sqrt{b \left ( ae-bd \right ) }xab{e}^{2}+35\,\sqrt{b \left ( ae-bd \right ) }x{b}^{2}de+3\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{2}-11\,\sqrt{b \left ( ae-bd \right ) }abde+23\,\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x)

[Out]

-2/15*(b*x+a)*(15*b^3*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)+
15*(b*(a*e-b*d))^(1/2)*x^2*b^2*e^2-5*(b*(a*e-b*d))^(1/2)*x*a*b*e^2+35*(b*(a*e-b*
d))^(1/2)*x*b^2*d*e+3*(b*(a*e-b*d))^(1/2)*a^2*e^2-11*(b*(a*e-b*d))^(1/2)*a*b*d*e
+23*(b*(a*e-b*d))^(1/2)*b^2*d^2)/((b*x+a)^2)^(1/2)/(a*e-b*d)^3/(e*x+d)^(5/2)/(b*
(a*e-b*d))^(1/2)

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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/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222821, size = 1, normalized size = 0. \[ \left [\frac{30 \, b^{2} e^{2} x^{2} + 46 \, b^{2} d^{2} - 22 \, a b d e + 6 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 10 \,{\left (7 \, b^{2} d e - a b e^{2}\right )} x}{15 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 11 \, a b d e + 3 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 5 \,{\left (7 \, b^{2} d e - a b e^{2}\right )} x\right )}}{15 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/15*(30*b^2*e^2*x^2 + 46*b^2*d^2 - 22*a*b*d*e + 6*a^2*e^2 - 15*(b^2*e^2*x^2 +
2*b^2*d*e*x + b^2*d^2)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*
e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 10*(7*b^2*d*e
- a*b*e^2)*x)/((b^3*d^5 - 3*a*b^2*d^4*e + 3*a^2*b*d^3*e^2 - a^3*d^2*e^3 + (b^3*d
^3*e^2 - 3*a*b^2*d^2*e^3 + 3*a^2*b*d*e^4 - a^3*e^5)*x^2 + 2*(b^3*d^4*e - 3*a*b^2
*d^3*e^2 + 3*a^2*b*d^2*e^3 - a^3*d*e^4)*x)*sqrt(e*x + d)), 2/15*(15*b^2*e^2*x^2
+ 23*b^2*d^2 - 11*a*b*d*e + 3*a^2*e^2 - 15*(b^2*e^2*x^2 + 2*b^2*d*e*x + b^2*d^2)
*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sq
rt(e*x + d)*b)) + 5*(7*b^2*d*e - a*b*e^2)*x)/((b^3*d^5 - 3*a*b^2*d^4*e + 3*a^2*b
*d^3*e^2 - a^3*d^2*e^3 + (b^3*d^3*e^2 - 3*a*b^2*d^2*e^3 + 3*a^2*b*d*e^4 - a^3*e^
5)*x^2 + 2*(b^3*d^4*e - 3*a*b^2*d^3*e^2 + 3*a^2*b*d^2*e^3 - a^3*d*e^4)*x)*sqrt(e
*x + d))]

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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/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218873, size = 265, normalized size = 1.21 \[ \frac{2}{15} \,{\left (\frac{15 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{2} b^{2} + 5 \,{\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} - 5 \,{\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{5}{2}}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

2/15*(15*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3 - 3*a*b^2*d^
2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) + (15*(x*e + d)^2*b^2 + 5*(
x*e + d)*b^2*d + 3*b^2*d^2 - 5*(x*e + d)*a*b*e - 6*a*b*d*e + 3*a^2*e^2)/((b^3*d^
3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(x*e + d)^(5/2)))*sign(b*x + a)