Optimal. Leaf size=157 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.257674, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[c*x]*(a + b*x^2)^(5/2)),x]
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Rubi in Sympy [A] time = 23.5794, size = 139, normalized size = 0.89 \[ \frac{\sqrt{c x}}{3 a c \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{5 \sqrt{c x}}{6 a^{2} c \sqrt{a + b x^{2}}} + \frac{5 \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{c} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(1/2)/(b*x**2+a)**(5/2),x)
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Mathematica [C] time = 0.333311, size = 115, normalized size = 0.73 \[ \frac{x \left (\frac{5 i \sqrt{x} \sqrt{\frac{a}{b x^2}+1} \left (a+b x^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+7 a+5 b x^2\right )}{6 a^2 \sqrt{c x} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(5/2)),x]
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Maple [A] time = 0.028, size = 216, normalized size = 1.4 \[{\frac{1}{12\,{a}^{2}b} \left ( 5\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}\sqrt{2}{x}^{2}b+5\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a+10\,{b}^{2}{x}^{3}+14\,abx \right ){\frac{1}{\sqrt{cx}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(1/2)/(b*x^2+a)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}} \sqrt{c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*sqrt(c*x)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{c x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*sqrt(c*x)),x, algorithm="fricas")
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Sympy [A] time = 54.9977, size = 44, normalized size = 0.28 \[ \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{c} \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(1/2)/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}} \sqrt{c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*sqrt(c*x)),x, algorithm="giac")
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