Optimal. Leaf size=153 \[ -\frac {5 a^{3/4} c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{6 b^{9/4} \sqrt {a+b x^2}}+\frac {5 c^3 \sqrt {c x} \sqrt {a+b x^2}}{3 b^2}-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {288, 321, 329, 220} \[ -\frac {5 a^{3/4} c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{6 b^{9/4} \sqrt {a+b x^2}}+\frac {5 c^3 \sqrt {c x} \sqrt {a+b x^2}}{3 b^2}-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 288
Rule 321
Rule 329
Rubi steps
\begin {align*} \int \frac {(c x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}}+\frac {\left (5 c^2\right ) \int \frac {(c x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{2 b}\\ &=-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}}+\frac {5 c^3 \sqrt {c x} \sqrt {a+b x^2}}{3 b^2}-\frac {\left (5 a c^4\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{6 b^2}\\ &=-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}}+\frac {5 c^3 \sqrt {c x} \sqrt {a+b x^2}}{3 b^2}-\frac {\left (5 a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{3 b^2}\\ &=-\frac {c (c x)^{5/2}}{b \sqrt {a+b x^2}}+\frac {5 c^3 \sqrt {c x} \sqrt {a+b x^2}}{3 b^2}-\frac {5 a^{3/4} 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 )}{6 b^{9/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 74, normalized size = 0.48 \[ \frac {c^3 \sqrt {c x} \left (-5 a \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )+5 a+2 b x^2\right )}{3 b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {c x} c^{3} x^{3}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 128, normalized size = 0.84 \[ -\frac {\sqrt {c x}\, \left (-4 b^{2} x^{3}-10 a b x +5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, a \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) c^{3}}{6 \sqrt {b \,x^{2}+a}\, b^{3} x} \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.73, size = 44, normalized size = 0.29 \[ \frac {c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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