3.997 \(\int \frac{1}{(c x)^{11/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=157 \[ \frac{16 b^{5/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} c^6 \sqrt [4]{a+b x^2}}-\frac{8 b^2}{3 a^3 c^5 \sqrt{c x} \sqrt [4]{a+b x^2}}+\frac{4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}} \]

[Out]

-2/(9*a*c*(c*x)^(9/2)*(a + b*x^2)^(1/4)) + (4*b)/(9*a^2*c^3*(c*x)^(5/2)*(a + b*x
^2)^(1/4)) - (8*b^2)/(3*a^3*c^5*Sqrt[c*x]*(a + b*x^2)^(1/4)) + (16*b^(5/2)*(1 +
a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*a^(7/
2)*c^6*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.215549, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{16 b^{5/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} c^6 \sqrt [4]{a+b x^2}}-\frac{8 b^2}{3 a^3 c^5 \sqrt{c x} \sqrt [4]{a+b x^2}}+\frac{4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(11/2)*(a + b*x^2)^(5/4)),x]

[Out]

-2/(9*a*c*(c*x)^(9/2)*(a + b*x^2)^(1/4)) + (4*b)/(9*a^2*c^3*(c*x)^(5/2)*(a + b*x
^2)^(1/4)) - (8*b^2)/(3*a^3*c^5*Sqrt[c*x]*(a + b*x^2)^(1/4)) + (16*b^(5/2)*(1 +
a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*a^(7/
2)*c^6*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2}{9 a c \left (c x\right )^{\frac{9}{2}} \sqrt [4]{a + b x^{2}}} + \frac{4 b}{9 a^{2} c^{3} \left (c x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}} - \frac{8 b^{2}}{3 a^{3} c^{5} \sqrt{c x} \sqrt [4]{a + b x^{2}}} + \frac{8 b^{2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{3 a^{3} c^{6} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(11/2)/(b*x**2+a)**(5/4),x)

[Out]

-2/(9*a*c*(c*x)**(9/2)*(a + b*x**2)**(1/4)) + 4*b/(9*a**2*c**3*(c*x)**(5/2)*(a +
b*x**2)**(1/4)) - 8*b**2/(3*a**3*c**5*sqrt(c*x)*(a + b*x**2)**(1/4)) + 8*b**2*s
qrt(c*x)*(a/(b*x**2) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-5/4), (x, 1/x))/(3*a
**3*c**6*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.122338, size = 105, normalized size = 0.67 \[ \frac{\sqrt{c x} \left (32 b^3 x^6 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-2 \left (a^3-2 a^2 b x^2+12 a b^2 x^4+24 b^3 x^6\right )\right )}{9 a^4 c^6 x^5 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(5/4)),x]

[Out]

(Sqrt[c*x]*(-2*(a^3 - 2*a^2*b*x^2 + 12*a*b^2*x^4 + 24*b^3*x^6) + 32*b^3*x^6*(1 +
(b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(9*a^4*c^6*x^
5*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{11}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(11/2)/(b*x^2+a)^(5/4),x)

[Out]

int(1/(c*x)^(11/2)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(11/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(11/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b c^{5} x^{7} + a c^{5} x^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(11/2)),x, algorithm="fricas")

[Out]

integral(1/((b*c^5*x^7 + a*c^5*x^5)*(b*x^2 + a)^(1/4)*sqrt(c*x)), 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(1/(c*x)**(11/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(11/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(11/2)), x)