Optimal. Leaf size=86 \[ \frac {\sqrt {a} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {321, 329, 237, 335, 275, 231} \[ \frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}+\frac {\sqrt {a} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 321
Rule 329
Rule 335
Rubi steps
\begin {align*} \int \frac {(c x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}-\frac {\left (a c^2\right ) \int \frac {1}{\sqrt {c x} \left (a+b x^2\right )^{3/4}} \, dx}{2 b}\\ &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{b}\\ &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}-\frac {\left (a c \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {c x}\right )}{b \left (a+b x^2\right )^{3/4}}\\ &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}+\frac {\left (a c \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{b \left (a+b x^2\right )^{3/4}}\\ &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}+\frac {\left (a c \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{c x}\right )}{2 b \left (a+b x^2\right )^{3/4}}\\ &=\frac {c \sqrt {c x} \sqrt [4]{a+b x^2}}{b}+\frac {\sqrt {a} \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 66, normalized size = 0.77 \[ \frac {c \sqrt {c x} \left (-a \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^2}{a}\right )+a+b x^2\right )}{b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x} c x}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {3}{2}}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.03, size = 44, normalized size = 0.51 \[ \frac {c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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