3.86 \(\int \frac{x^{7/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^3}}\right )}{a^{9/2}}+\frac{\sqrt{x}}{a^4 \sqrt{a x+b x^3}}+\frac{x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

[Out]

x^(7/2)/(7*a*(a*x + b*x^3)^(7/2)) + x^(5/2)/(5*a^2*(a*x + b*x^3)^(5/2)) + x^(3/2
)/(3*a^3*(a*x + b*x^3)^(3/2)) + Sqrt[x]/(a^4*Sqrt[a*x + b*x^3]) - ArcTanh[(Sqrt[
a]*Sqrt[x])/Sqrt[a*x + b*x^3]]/a^(9/2)

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Rubi [A]  time = 0.32764, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^3}}\right )}{a^{9/2}}+\frac{\sqrt{x}}{a^4 \sqrt{a x+b x^3}}+\frac{x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(7/2)/(7*a*(a*x + b*x^3)^(7/2)) + x^(5/2)/(5*a^2*(a*x + b*x^3)^(5/2)) + x^(3/2
)/(3*a^3*(a*x + b*x^3)^(3/2)) + Sqrt[x]/(a^4*Sqrt[a*x + b*x^3]) - ArcTanh[(Sqrt[
a]*Sqrt[x])/Sqrt[a*x + b*x^3]]/a^(9/2)

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Rubi in Sympy [A]  time = 34.4343, size = 112, normalized size = 0.86 \[ \frac{x^{\frac{7}{2}}}{7 a \left (a x + b x^{3}\right )^{\frac{7}{2}}} + \frac{x^{\frac{5}{2}}}{5 a^{2} \left (a x + b x^{3}\right )^{\frac{5}{2}}} + \frac{x^{\frac{3}{2}}}{3 a^{3} \left (a x + b x^{3}\right )^{\frac{3}{2}}} + \frac{\sqrt{x}}{a^{4} \sqrt{a x + b x^{3}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{3}}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(b*x**3+a*x)**(9/2),x)

[Out]

x**(7/2)/(7*a*(a*x + b*x**3)**(7/2)) + x**(5/2)/(5*a**2*(a*x + b*x**3)**(5/2)) +
x**(3/2)/(3*a**3*(a*x + b*x**3)**(3/2)) + sqrt(x)/(a**4*sqrt(a*x + b*x**3)) - a
tanh(sqrt(a)*sqrt(x)/sqrt(a*x + b*x**3))/a**(9/2)

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Mathematica [A]  time = 0.200504, size = 123, normalized size = 0.95 \[ \frac{\sqrt{x} \left (\sqrt{a} \left (176 a^3+406 a^2 b x^2+350 a b^2 x^4+105 b^3 x^6\right )+105 \log (x) \left (a+b x^2\right )^{7/2}-105 \left (a+b x^2\right )^{7/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )\right )}{105 a^{9/2} \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a*x + b*x^3)^(9/2),x]

[Out]

(Sqrt[x]*(Sqrt[a]*(176*a^3 + 406*a^2*b*x^2 + 350*a*b^2*x^4 + 105*b^3*x^6) + 105*
(a + b*x^2)^(7/2)*Log[x] - 105*(a + b*x^2)^(7/2)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]
]))/(105*a^(9/2)*(a + b*x^2)^3*Sqrt[x*(a + b*x^2)])

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Maple [B]  time = 0.017, size = 217, normalized size = 1.7 \[ -{\frac{1}{105\, \left ( b{x}^{2}+a \right ) ^{4}}\sqrt{x \left ( b{x}^{2}+a \right ) } \left ( 105\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ){x}^{6}{b}^{3}\sqrt{b{x}^{2}+a}-105\,\sqrt{a}{x}^{6}{b}^{3}+315\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ){x}^{4}a{b}^{2}\sqrt{b{x}^{2}+a}-350\,{a}^{3/2}{x}^{4}{b}^{2}+315\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ){x}^{2}{a}^{2}b\sqrt{b{x}^{2}+a}-406\,{x}^{2}b{a}^{5/2}+105\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ){a}^{3}\sqrt{b{x}^{2}+a}-176\,{a}^{7/2} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(b*x^3+a*x)^(9/2),x)

[Out]

-1/105*(x*(b*x^2+a))^(1/2)/a^(9/2)*(105*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*x^6*
b^3*(b*x^2+a)^(1/2)-105*a^(1/2)*x^6*b^3+315*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*
x^4*a*b^2*(b*x^2+a)^(1/2)-350*a^(3/2)*x^4*b^2+315*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+
a)/x)*x^2*a^2*b*(b*x^2+a)^(1/2)-406*x^2*b*a^(5/2)+105*ln(2*(a^(1/2)*(b*x^2+a)^(1
/2)+a)/x)*a^3*(b*x^2+a)^(1/2)-176*a^(7/2))/x^(1/2)/(b*x^2+a)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{7}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x^3 + a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(7/2)/(b*x^3 + a*x)^(9/2), x)

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Fricas [A]  time = 0.224973, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{3} + a x} \sqrt{x} \log \left (-\frac{2 \, \sqrt{b x^{3} + a x} a \sqrt{x} -{\left (b x^{3} + 2 \, a x\right )} \sqrt{a}}{x^{3}}\right ) + 2 \,{\left (105 \, b^{3} x^{7} + 350 \, a b^{2} x^{5} + 406 \, a^{2} b x^{3} + 176 \, a^{3} x\right )} \sqrt{a}}{210 \,{\left (a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} + 3 \, a^{6} b x^{2} + a^{7}\right )} \sqrt{b x^{3} + a x} \sqrt{a} \sqrt{x}}, -\frac{105 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{3} + a x} \sqrt{x} \arctan \left (\frac{\sqrt{b x^{3} + a x} \sqrt{-a}}{a \sqrt{x}}\right ) -{\left (105 \, b^{3} x^{7} + 350 \, a b^{2} x^{5} + 406 \, a^{2} b x^{3} + 176 \, a^{3} x\right )} \sqrt{-a}}{105 \,{\left (a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} + 3 \, a^{6} b x^{2} + a^{7}\right )} \sqrt{b x^{3} + a x} \sqrt{-a} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x^3 + a*x)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^3 + a*x)*sqrt(x
)*log(-(2*sqrt(b*x^3 + a*x)*a*sqrt(x) - (b*x^3 + 2*a*x)*sqrt(a))/x^3) + 2*(105*b
^3*x^7 + 350*a*b^2*x^5 + 406*a^2*b*x^3 + 176*a^3*x)*sqrt(a))/((a^4*b^3*x^6 + 3*a
^5*b^2*x^4 + 3*a^6*b*x^2 + a^7)*sqrt(b*x^3 + a*x)*sqrt(a)*sqrt(x)), -1/105*(105*
(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^3 + a*x)*sqrt(x)*arctan(sqr
t(b*x^3 + a*x)*sqrt(-a)/(a*sqrt(x))) - (105*b^3*x^7 + 350*a*b^2*x^5 + 406*a^2*b*
x^3 + 176*a^3*x)*sqrt(-a))/((a^4*b^3*x^6 + 3*a^5*b^2*x^4 + 3*a^6*b*x^2 + a^7)*sq
rt(b*x^3 + a*x)*sqrt(-a)*sqrt(x))]

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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(x**(7/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230585, size = 154, normalized size = 1.18 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{105 \, \sqrt{a} \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + 176 \, \sqrt{-a}}{105 \, \sqrt{-a} a^{\frac{9}{2}}} + \frac{105 \,{\left (b x^{2} + a\right )}^{3} + 35 \,{\left (b x^{2} + a\right )}^{2} a + 21 \,{\left (b x^{2} + a\right )} a^{2} + 15 \, a^{3}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x^3 + a*x)^(9/2),x, algorithm="giac")

[Out]

arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) - 1/105*(105*sqrt(a)*arctan(sqrt
(a)/sqrt(-a)) + 176*sqrt(-a))/(sqrt(-a)*a^(9/2)) + 1/105*(105*(b*x^2 + a)^3 + 35
*(b*x^2 + a)^2*a + 21*(b*x^2 + a)*a^2 + 15*a^3)/((b*x^2 + a)^(7/2)*a^4)