Optimal. Leaf size=157 \[ \frac {80 b^{7/2} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{77 a^{7/2} c^8 \left (a+b x^2\right )^{3/4}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac {80 b^{7/2} (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} c^8 \left (a+b x^2\right )^{3/4}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}} \]
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)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}-\frac {(10 b) \int \frac {1}{(c x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx}{11 a c^2}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}+\frac {\left (60 b^2\right ) \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{77 a^2 c^4}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}-\frac {\left (40 b^3\right ) \int \frac {1}{\sqrt {c x} \left (a+b x^2\right )^{3/4}} \, dx}{77 a^3 c^6}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}-\frac {\left (80 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{77 a^3 c^7}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}-\frac {\left (80 b^3 \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 )}{77 a^3 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac {\left (80 b^3 \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 )}{77 a^3 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac {\left (40 b^3 \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 )}{77 a^3 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{11 a c (c x)^{11/2}}+\frac {20 b \sqrt [4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac {40 b^2 \sqrt [4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac {80 b^{7/2} \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 )}{77 a^{7/2} c^8 \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 56, normalized size = 0.36 \[ -\frac {2 x \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {11}{4},\frac {3}{4};-\frac {7}{4};-\frac {b x^2}{a}\right )}{11 (c x)^{13/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{b c^{7} x^{9} + a c^{7} x^{7}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {13}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\left (c x \right )^{\frac {13}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {13}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{13/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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