Optimal. Leaf size=118 \[ -\frac {3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{7/2}}+\frac {3 d (4 b c-5 a d)}{8 c^3 \sqrt {c+\frac {d}{x^2}}}+\frac {x^2 (4 b c-5 a d)}{8 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}} \]
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Rubi [A] time = 0.08, antiderivative size = 120, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \begin {gather*} \frac {3 x^2 \sqrt {c+\frac {d}{x^2}} (4 b c-5 a d)}{8 c^3}-\frac {x^2 (4 b c-5 a d)}{4 c^2 \sqrt {c+\frac {d}{x^2}}}-\frac {3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{7/2}}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) x^3}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{x^3 (c+d x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}}-\frac {\left (2 b c-\frac {5 a d}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (c+d x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )}{4 c}\\ &=-\frac {(4 b c-5 a d) x^2}{4 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}}-\frac {(3 (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{8 c^2}\\ &=-\frac {(4 b c-5 a d) x^2}{4 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {3 (4 b c-5 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^3}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}}+\frac {(3 d (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{16 c^3}\\ &=-\frac {(4 b c-5 a d) x^2}{4 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {3 (4 b c-5 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^3}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}}+\frac {(3 (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{8 c^3}\\ &=-\frac {(4 b c-5 a d) x^2}{4 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {3 (4 b c-5 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^3}+\frac {a x^4}{4 c \sqrt {c+\frac {d}{x^2}}}-\frac {3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 111, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c} x \left (a \left (2 c^2 x^4-5 c d x^2-15 d^2\right )+4 b c \left (c x^2+3 d\right )\right )+3 d^{3/2} \sqrt {\frac {c x^2}{d}+1} (5 a d-4 b c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{8 c^{7/2} x \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 119, normalized size = 1.01 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (2 a c^2 x^6-5 a c d x^4-15 a d^2 x^2+4 b c^2 x^4+12 b c d x^2\right )}{8 c^3 \left (c x^2+d\right )}-\frac {3 \left (4 b c d-5 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )}{8 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 304, normalized size = 2.58 \begin {gather*} \left [-\frac {3 \, {\left (4 \, b c d^{2} - 5 \, a d^{3} + {\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (2 \, a c^{3} x^{6} + {\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \, {\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, {\left (c^{5} x^{2} + c^{4} d\right )}}, \frac {3 \, {\left (4 \, b c d^{2} - 5 \, a d^{3} + {\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (2 \, a c^{3} x^{6} + {\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \, {\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, {\left (c^{5} x^{2} + c^{4} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 140, normalized size = 1.19 \begin {gather*} \frac {\left (c \,x^{2}+d \right ) \left (2 a \,c^{\frac {7}{2}} x^{5}-5 a \,c^{\frac {5}{2}} d \,x^{3}+4 b \,c^{\frac {7}{2}} x^{3}-15 a \,c^{\frac {3}{2}} d^{2} x +12 b \,c^{\frac {5}{2}} d x +15 \sqrt {c \,x^{2}+d}\, a c \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-12 \sqrt {c \,x^{2}+d}\, b \,c^{2} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )\right )}{8 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} c^{\frac {9}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 215, normalized size = 1.82 \begin {gather*} -\frac {1}{16} \, a {\left (\frac {2 \, {\left (15 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} - 25 \, {\left (c + \frac {d}{x^{2}}\right )} c d^{2} + 8 \, c^{2} d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3} - 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{4} + \sqrt {c + \frac {d}{x^{2}}} c^{5}} + \frac {15 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} + \frac {1}{4} \, b {\left (\frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )} d - 2 \, c d\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} - \sqrt {c + \frac {d}{x^{2}}} c^{3}} + \frac {3 \, d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.34, size = 134, normalized size = 1.14 \begin {gather*} \frac {a\,x^4}{4\,c\,\sqrt {c+\frac {d}{x^2}}}-\frac {15\,a\,d^2}{8\,c^3\,\sqrt {c+\frac {d}{x^2}}}+\frac {b\,x^2}{2\,c\,\sqrt {c+\frac {d}{x^2}}}-\frac {3\,b\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{5/2}}+\frac {15\,a\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,c^{7/2}}+\frac {3\,b\,d}{2\,c^2\,\sqrt {c+\frac {d}{x^2}}}-\frac {5\,a\,d\,x^2}{8\,c^2\,\sqrt {c+\frac {d}{x^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 103.90, size = 177, normalized size = 1.50 \begin {gather*} a \left (\frac {x^{5}}{4 c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {5 \sqrt {d} x^{3}}{8 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {15 d^{\frac {3}{2}} x}{8 c^{3} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {15 d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 c^{\frac {7}{2}}}\right ) + b \left (\frac {x^{3}}{2 c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 \sqrt {d} x}{2 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {3 d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {5}{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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