Optimal. Leaf size=185 \[ -\frac {5 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{4 a^{13/4} c^{5/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {290, 325, 329, 220} \[ -\frac {5 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{4 a^{13/4} c^{5/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 325
Rule 329
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3 \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx}{2 a}\\ &=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}+\frac {15 \int \frac {1}{(c x)^{5/2} \sqrt {a+b x^2}} \, dx}{4 a^2}\\ &=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac {(5 b) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{4 a^3 c^2}\\ &=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{2 a^3 c^3}\\ &=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac {5 b^{3/4} \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 )}{4 a^{13/4} c^{5/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.32 \[ -\frac {2 x \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {5}{2};\frac {1}{4};-\frac {b x^2}{a}\right )}{3 a^2 (c x)^{5/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {c x}}{b^{3} c^{3} x^{9} + 3 \, a b^{2} c^{3} x^{7} + 3 \, a^{2} b c^{3} x^{5} + a^{3} c^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 227, normalized size = 1.23 \[ -\frac {30 b^{2} x^{4}+15 \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, b \,x^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+42 a b \,x^{2}+15 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, a x \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+8 a^{2}}{12 \sqrt {c x}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3} c^{2} x} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.97, size = 48, normalized size = 0.26 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} c^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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