Optimal. Leaf size=244 \[ -\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}+\frac {\sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac {d^4 \left (5 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5 \left (a e^2+c d^2\right )^{3/2}}-\frac {5 d \sqrt {a+c x^2} (d+e x)}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^4} \]
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Rubi [A] time = 0.89, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1651, 1654, 844, 217, 206, 725} \[ \frac {\sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}-\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac {d^4 \left (5 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5 \left (a e^2+c d^2\right )^{3/2}}-\frac {5 d \sqrt {a+c x^2} (d+e x)}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1651
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {-\frac {a d^4}{e^3}+\frac {d^3 \left (c d^2+a e^2\right ) x}{e^4}-\frac {d^2 \left (c d^2+a e^2\right ) x^2}{e^3}+d \left (a+\frac {c d^2}{e^2}\right ) x^3-\frac {\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2}\\ &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-a d^2 e \left (c d^2-2 a e^2\right )+4 d \left (c d^2+a e^2\right )^2 x+2 e \left (c d^2+a e^2\right )^2 x^2+10 c d e^2 \left (c d^2+a e^2\right ) x^3}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 c e^4 \left (c d^2+a e^2\right )}\\ &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c d^2 e^4 \left (2 c d^2+a e^2\right )-2 c d e^3 \left (c d^2+a e^2\right )^2 x-2 c e^4 \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^2 e^7 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c^2 d^2 e^6 \left (2 c d^2+a e^2\right )+6 c^2 d e^5 \left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^3 e^9 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c e^5}+\frac {\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e^5}-\frac {\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \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^5 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 230, normalized size = 0.94 \[ \frac {-\frac {3 d \left (4 c d^2-a e^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{c^{3/2}}+e \sqrt {a+c x^2} \left (-\frac {2 a e^2}{c^2}+\frac {3 d^5}{(d+e x) \left (a e^2+c d^2\right )}+\frac {9 d^2-3 d e x+e^2 x^2}{c}\right )-\frac {3 d^4 \left (5 a e^2+4 c d^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac {3 d^4 \left (5 a e^2+4 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{3 e^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 91.20, size = 2025, normalized size = 8.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 474, normalized size = 1.94 \[ \frac {c \,d^{6} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{6}}+\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{5}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) e^{5}}+\frac {\sqrt {c \,x^{2}+a}\, x^{2}}{3 c \,e^{2}}-\frac {5 d^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{6}}+\frac {a d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}} e^{3}}-\frac {4 d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}\, e^{5}}-\frac {\sqrt {c \,x^{2}+a}\, d x}{c \,e^{3}}-\frac {2 \sqrt {c \,x^{2}+a}\, a}{3 c^{2} e^{2}}+\frac {3 \sqrt {c \,x^{2}+a}\, d^{2}}{c \,e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 274, normalized size = 1.12 \[ \frac {\sqrt {c x^{2} + a} d^{5}}{c d^{2} e^{5} x + a e^{7} x + c d^{3} e^{4} + a d e^{6}} + \frac {\sqrt {c x^{2} + a} x^{2}}{3 \, c e^{2}} - \frac {\sqrt {c x^{2} + a} d x}{c e^{3}} - \frac {4 \, d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} e^{5}} + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}} e^{3}} - \frac {c d^{6} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{8}} + \frac {5 \, d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{\sqrt {a + \frac {c d^{2}}{e^{2}}} e^{6}} + \frac {3 \, \sqrt {c x^{2} + a} d^{2}}{c e^{4}} - \frac {2 \, \sqrt {c x^{2} + a} a}{3 \, c^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{5}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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