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

Optimal. Leaf size=90 \[ \frac{3 \sqrt{a} c^2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^2}}+\frac{c (c x)^{3/2}}{b \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.114394, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 \sqrt{a} c^2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^2}}+\frac{c (c x)^{3/2}}{b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0578222, size = 60, normalized size = 0.67 \[ \frac{2 c (c x)^{3/2} \left (\sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-1\right )}{b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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