Optimal. Leaf size=126 \[ -\frac {8 b^{5/2} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{7 a^{5/2} c^6 \left (a+b x^2\right )^{3/4}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {325, 329, 237, 335, 275, 231} \[ -\frac {8 b^{5/2} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a+b x^2\right )^{3/4}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 325
Rule 329
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}-\frac {(6 b) \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{7 a c^2}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (4 b^2\right ) \int \frac {1}{\sqrt {c x} \left (a+b x^2\right )^{3/4}} \, dx}{7 a^2 c^4}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{7 a^2 c^5}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (8 b^2 \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 )}{7 a^2 c^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {\left (8 b^2 \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 )}{7 a^2 c^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {\left (4 b^2 \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 )}{7 a^2 c^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{7 a c (c x)^{7/2}}+\frac {4 b \sqrt [4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {8 b^{5/2} \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 )}{7 a^{5/2} c^6 \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 56, normalized size = 0.44 \[ -\frac {2 x \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {7}{4},\frac {3}{4};-\frac {3}{4};-\frac {b x^2}{a}\right )}{7 (c x)^{9/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{b c^{5} x^{7} + a c^{5} x^{5}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{9/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 66.19, size = 34, normalized size = 0.27 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 b^{\frac {3}{4}} c^{\frac {9}{2}} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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