3.953 \(\int \frac{1}{(c x)^{7/2} \sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a+b x^2}}+\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.158869, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a+b x^2}}+\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(7/2)*(a + b*x^2)^(1/4)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(7/2)/(b*x**2+a)**(1/4),x)

[Out]

4*b/(5*a*c**3*sqrt(c*x)*(a + b*x**2)**(1/4)) + 2*b*sqrt(c*x)*(a/(b*x**2) + 1)**(
1/4)*Integral((a*x**2/b + 1)**(-1/4), (x, 1/x))/(5*a*c**4*(a + b*x**2)**(1/4)) -
4*b*sqrt(c*x)/(5*a*c**4*x*(a + b*x**2)**(1/4)) - 2*(a + b*x**2)**(3/4)/(5*a*c*(
c*x)**(5/2))

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Mathematica [C]  time = 0.0714954, size = 88, normalized size = 0.7 \[ \frac{x \left (-6 a^2-8 b^2 x^4 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+6 a b x^2+12 b^2 x^4\right )}{15 a^2 (c x)^{7/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(1/4)),x]

[Out]

(x*(-6*a^2 + 6*a*b*x^2 + 12*b^2*x^4 - 8*b^2*x^4*(1 + (b*x^2)/a)^(1/4)*Hypergeome
tric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(15*a^2*(c*x)^(7/2)*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(7/2)/(b*x^2+a)^(1/4),x)

[Out]

int(1/(c*x)^(7/2)/(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(1/4)*sqrt(c*x)*c^3*x^3), 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)**(7/2)/(b*x**2+a)**(1/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(7/2)), x)