Optimal. Leaf size=155 \[ \frac {5 c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {288, 329, 220} \[ -\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {5 c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}}-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 288
Rule 329
Rubi steps
\begin {align*} \int \frac {(c x)^{7/2}}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {(c x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{6 b}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {\left (5 c^4\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{12 b^2}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{6 b^2}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {5 c^{7/2} \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 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 80, normalized size = 0.52 \[ \frac {c^3 \sqrt {c x} \left (5 \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )-5 a-7 b x^2\right )}{6 b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {c x} c^{3} x^{3}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 219, normalized size = 1.41 \[ \frac {\left (-14 b^{2} x^{3}+5 \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, b \,x^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-10 a b x +5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, a \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {c x}\, c^{3}}{12 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3} x} \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.87, size = 44, normalized size = 0.28 \[ \frac {c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {5}{2} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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