3.358 \(\int \frac{1}{x^3 (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 b \sqrt{a+b x}}{4 a^4 x}-\frac{35 \sqrt{a+b x}}{6 a^3 x^2}+\frac{14}{3 a^2 x^2 \sqrt{a+b x}}+\frac{2}{3 a x^2 (a+b x)^{3/2}} \]

[Out]

2/(3*a*x^2*(a + b*x)^(3/2)) + 14/(3*a^2*x^2*Sqrt[a + b*x]) - (35*Sqrt[a + b*x])/
(6*a^3*x^2) + (35*b*Sqrt[a + b*x])/(4*a^4*x) - (35*b^2*ArcTanh[Sqrt[a + b*x]/Sqr
t[a]])/(4*a^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0997253, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 b \sqrt{a+b x}}{4 a^4 x}-\frac{35 \sqrt{a+b x}}{6 a^3 x^2}+\frac{14}{3 a^2 x^2 \sqrt{a+b x}}+\frac{2}{3 a x^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^(5/2)),x]

[Out]

2/(3*a*x^2*(a + b*x)^(3/2)) + 14/(3*a^2*x^2*Sqrt[a + b*x]) - (35*Sqrt[a + b*x])/
(6*a^3*x^2) + (35*b*Sqrt[a + b*x])/(4*a^4*x) - (35*b^2*ArcTanh[Sqrt[a + b*x]/Sqr
t[a]])/(4*a^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.0794, size = 99, normalized size = 0.93 \[ \frac{2}{3 a x^{2} \left (a + b x\right )^{\frac{3}{2}}} + \frac{14}{3 a^{2} x^{2} \sqrt{a + b x}} - \frac{35 \sqrt{a + b x}}{6 a^{3} x^{2}} + \frac{35 b \sqrt{a + b x}}{4 a^{4} x} - \frac{35 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x+a)**(5/2),x)

[Out]

2/(3*a*x**2*(a + b*x)**(3/2)) + 14/(3*a**2*x**2*sqrt(a + b*x)) - 35*sqrt(a + b*x
)/(6*a**3*x**2) + 35*b*sqrt(a + b*x)/(4*a**4*x) - 35*b**2*atanh(sqrt(a + b*x)/sq
rt(a))/(4*a**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.136505, size = 78, normalized size = 0.74 \[ \frac{-6 a^3+21 a^2 b x+140 a b^2 x^2+105 b^3 x^3}{12 a^4 x^2 (a+b x)^{3/2}}-\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^(5/2)),x]

[Out]

(-6*a^3 + 21*a^2*b*x + 140*a*b^2*x^2 + 105*b^3*x^3)/(12*a^4*x^2*(a + b*x)^(3/2))
- (35*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 80, normalized size = 0.8 \[ 2\,{b}^{2} \left ( 3\,{\frac{1}{{a}^{4}\sqrt{bx+a}}}+1/3\,{\frac{1}{{a}^{3} \left ( bx+a \right ) ^{3/2}}}+{\frac{1}{{a}^{4}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ({\frac{11\, \left ( bx+a \right ) ^{3/2}}{8}}-{\frac{13\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{35}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(5/2),x)

[Out]

2*b^2*(3/a^4/(b*x+a)^(1/2)+1/3/a^3/(b*x+a)^(3/2)+1/a^4*((11/8*(b*x+a)^(3/2)-13/8
*a*(b*x+a)^(1/2))/x^2/b^2-35/8*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.227876, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (105 \, b^{3} x^{3} + 140 \, a b^{2} x^{2} + 21 \, a^{2} b x - 6 \, a^{3}\right )} \sqrt{a}}{24 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{a}}, \frac{105 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (105 \, b^{3} x^{3} + 140 \, a b^{2} x^{2} + 21 \, a^{2} b x - 6 \, a^{3}\right )} \sqrt{-a}}{12 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[1/24*(105*(b^3*x^3 + a*b^2*x^2)*sqrt(b*x + a)*log(((b*x + 2*a)*sqrt(a) - 2*sqrt
(b*x + a)*a)/x) + 2*(105*b^3*x^3 + 140*a*b^2*x^2 + 21*a^2*b*x - 6*a^3)*sqrt(a))/
((a^4*b*x^3 + a^5*x^2)*sqrt(b*x + a)*sqrt(a)), 1/12*(105*(b^3*x^3 + a*b^2*x^2)*s
qrt(b*x + a)*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (105*b^3*x^3 + 140*a*b^2*x^2 +
21*a^2*b*x - 6*a^3)*sqrt(-a))/((a^4*b*x^3 + a^5*x^2)*sqrt(b*x + a)*sqrt(-a))]

_______________________________________________________________________________________

Sympy [A]  time = 27.2254, size = 464, normalized size = 4.38 \[ - \frac{6 a^{\frac{89}{2}} b^{75} x^{75}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{21 a^{\frac{87}{2}} b^{76} x^{76}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{140 a^{\frac{85}{2}} b^{77} x^{77}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{105 a^{\frac{83}{2}} b^{78} x^{78}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{105 a^{42} b^{\frac{155}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{105 a^{41} b^{\frac{157}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a)**(5/2),x)

[Out]

-6*a**(89/2)*b**75*x**75/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) +
12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1)) + 21*a**(87/2)*b**76*x**7
6/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2
)*x**(157/2)*sqrt(a/(b*x) + 1)) + 140*a**(85/2)*b**77*x**77/(12*a**(93/2)*b**(15
1/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b
*x) + 1)) + 105*a**(83/2)*b**78*x**78/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a
/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1)) - 105*a**42*
b**(155/2)*x**(155/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*a**
(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(157
/2)*sqrt(a/(b*x) + 1)) - 105*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) + 1)*asinh
(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1
) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205667, size = 126, normalized size = 1.19 \[ \frac{35 \, b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{4}} + \frac{2 \,{\left (9 \,{\left (b x + a\right )} b^{2} + a b^{2}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}} + \frac{11 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} - 13 \, \sqrt{b x + a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

35/4*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + 2/3*(9*(b*x + a)*b^2 +
a*b^2)/((b*x + a)^(3/2)*a^4) + 1/4*(11*(b*x + a)^(3/2)*b^2 - 13*sqrt(b*x + a)*a*
b^2)/(a^4*b^2*x^2)