3.992 \(\int \frac{(c x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{7 a^{3/2} c^4 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{7 a c^3 (c x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{7/2}}{3 b \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.15563, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{7 a^{3/2} c^4 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{7 a c^3 (c x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{c (c x)^{7/2}}{3 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(9/2)/(a + b*x^2)^(5/4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{7 a^{2} c^{4} \sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{4 b^{3} \sqrt [4]{a + b x^{2}}} - \frac{7 a^{2} c^{4} \sqrt{c x}}{2 b^{3} x \sqrt [4]{a + b x^{2}}} - \frac{7 a c^{3} \left (c x\right )^{\frac{3}{2}}}{6 b^{2} \sqrt [4]{a + b x^{2}}} + \frac{c \left (c x\right )^{\frac{7}{2}}}{3 b \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(9/2)/(b*x**2+a)**(5/4),x)

[Out]

7*a**2*c**4*sqrt(c*x)*(a/(b*x**2) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-1/4), (
x, 1/x))/(4*b**3*(a + b*x**2)**(1/4)) - 7*a**2*c**4*sqrt(c*x)/(2*b**3*x*(a + b*x
**2)**(1/4)) - 7*a*c**3*(c*x)**(3/2)/(6*b**2*(a + b*x**2)**(1/4)) + c*(c*x)**(7/
2)/(3*b*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.069475, size = 73, normalized size = 0.59 \[ \frac{c^3 (c x)^{3/2} \left (-7 a \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+7 a+b x^2\right )}{3 b^2 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(9/2)/(a + b*x^2)^(5/4),x]

[Out]

(c^3*(c*x)^(3/2)*(7*a + b*x^2 - 7*a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4,
3/4, 7/4, -((b*x^2)/a)]))/(3*b^2*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{9}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(9/2)/(b*x^2+a)^(5/4),x)

[Out]

int((c*x)^(9/2)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{9}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(9/2)/(b*x^2 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((c*x)^(9/2)/(b*x^2 + a)^(5/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(9/2)/(b*x^2 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(sqrt(c*x)*c^4*x^4/(b*x^2 + a)^(5/4), 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((c*x)**(9/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{9}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(9/2)/(b*x^2 + a)^(5/4),x, algorithm="giac")

[Out]

integrate((c*x)^(9/2)/(b*x^2 + a)^(5/4), x)