Optimal. Leaf size=146 \[ -\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}+\frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1647, 1654, 844, 217, 206, 725} \[ \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1647
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\int \frac {\frac {a^2 d^2}{c d^2+a e^2}-a x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a c}\\ &=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {\int \frac {\frac {a^2 c d^2 e^2}{c d^2+a e^2}+a c d e x}{(d+e x) \sqrt {a+c x^2}} \, dx}{a c^2 e^2}\\ &=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c e^2}+\frac {d^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^2 \left (c d^2+a e^2\right )}\\ &=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e^2}-\frac {d^4 \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^2 \left (c d^2+a e^2\right )}\\ &=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \left (c d^2+a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 179, normalized size = 1.23 \[ \frac {\frac {e \left (2 a^2 e^2+a c \left (d^2+d e x+e^2 x^2\right )+c^2 d^2 x^2\right )}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\sqrt {a} d \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2} \sqrt {a+c x^2}}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}}{e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 9.36, size = 1525, normalized size = 10.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 299, normalized size = 2.05 \[ \frac {2 \, d^{4} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e^{2} + a e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {d e^{\left (-2\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {3}{2}}} + \frac {{\left (\frac {{\left (c^{4} d^{4} e^{5} + 2 \, a c^{3} d^{2} e^{7} + a^{2} c^{2} e^{9}\right )} x}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}} + \frac {a c^{3} d^{3} e^{6} + a^{2} c^{2} d e^{8}}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}}\right )} x + \frac {a c^{3} d^{4} e^{5} + 3 \, a^{2} c^{2} d^{2} e^{7} + 2 \, a^{3} c e^{9}}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}}}{\sqrt {c x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 396, normalized size = 2.71 \[ \frac {c \,d^{5} x}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a \,e^{4}}-\frac {d^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{3}}+\frac {d^{4}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{3}}+\frac {x^{2}}{\sqrt {c \,x^{2}+a}\, c e}-\frac {d^{3} x}{\sqrt {c \,x^{2}+a}\, a \,e^{4}}+\frac {d x}{\sqrt {c \,x^{2}+a}\, c \,e^{2}}-\frac {d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}} e^{2}}+\frac {2 a}{\sqrt {c \,x^{2}+a}\, c^{2} e}-\frac {d^{2}}{\sqrt {c \,x^{2}+a}\, c \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 251, normalized size = 1.72 \[ \frac {c d^{5} x}{\sqrt {c x^{2} + a} a c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{2} e^{6}} + \frac {d^{4}}{\sqrt {c x^{2} + a} c d^{2} e^{3} + \sqrt {c x^{2} + a} a e^{5}} + \frac {x^{2}}{\sqrt {c x^{2} + a} c e} - \frac {d^{3} x}{\sqrt {c x^{2} + a} a e^{4}} + \frac {d x}{\sqrt {c x^{2} + a} c e^{2}} - \frac {d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}} e^{2}} + \frac {d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{5}} - \frac {d^{2}}{\sqrt {c x^{2} + a} c e^{3}} + \frac {2 \, a}{\sqrt {c x^{2} + a} c^{2} e} \]
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 \[ \int \frac {x^4}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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