Optimal. Leaf size=460 \[ -\frac {2 d^2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{e^3}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\right )}{e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 e^3}-\frac {b d \sqrt {x}}{c e^2}+\frac {b x^{3/2}}{6 c e} \]
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Rubi [A] time = 0.78, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {43, 5980, 5916, 302, 206, 321, 6044, 5920, 2402, 2315, 2447} \[ \frac {b d^2 \text {PolyLog}\left (2,1-\frac {2}{c \sqrt {x}+1}\right )}{e^3}-\frac {b d^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 e^3}-\frac {b d^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 e^3}-\frac {2 d^2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{e^3}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}+\frac {b \sqrt {x}}{2 c^3 e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {b d \sqrt {x}}{c e^2}+\frac {b x^{3/2}}{6 c e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 2447
Rule 5916
Rule 5920
Rule 5980
Rule 6044
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )}{e}\\ &=\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {(2 d) \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{e^2}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )}{e^2}-\frac {(b c) \operatorname {Subst}\left (\int \frac {x^4}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 e}\\ &=-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}+\frac {(b c d) \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^2}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {a+b \tanh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \tanh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{e^2}-\frac {(b c) \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 e}\\ &=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {d^2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{e^{5/2}}+\frac {d^2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{e^{5/2}}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c e^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 c^3 e}\\ &=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {2 d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+2 \frac {\left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^3}-\frac {\left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^3}-\frac {\left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^3}\\ &=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {2 d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}+2 \frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c \sqrt {x}}\right )}{e^3}\\ &=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {d x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {2 d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{1+c \sqrt {x}}\right )}{e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}\\ \end {align*}
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Mathematica [C] time = 2.93, size = 558, normalized size = 1.21 \[ \frac {6 a d^2 \log (d+e x)-6 a d e x+3 a e^2 x^2+\frac {b \left (-6 c^4 d^2 \left (\tanh ^{-1}\left (c \sqrt {x}\right ) \left (\tanh ^{-1}\left (c \sqrt {x}\right )+2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )\right )-\text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+2 c e \sqrt {x} \left (2 e-3 c^2 d\right )-6 e \left (c^2 x-1\right ) \left (c^2 d-e\right ) \tanh ^{-1}\left (c \sqrt {x}\right )+c e^2 \sqrt {x} \left (c^2 x-1\right )+3 e^2 \left (c^2 x-1\right )^2 \tanh ^{-1}\left (c \sqrt {x}\right )+3 c^4 d^2 \left (-\text {Li}_2\left (\frac {\left (-d c^2+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2+e}\right )-\text {Li}_2\left (\frac {\left (-d c^2+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2+e}\right )-4 i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \tanh ^{-1}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 \log \left (\frac {e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}-1\right )\right )}{c^2 d+e}\right ) \left (\tanh ^{-1}\left (c \sqrt {x}\right )-i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )\right )+2 \log \left (\frac {e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}-1\right )\right )}{c^2 d+e}\right ) \left (\tanh ^{-1}\left (c \sqrt {x}\right )+i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )\right )+2 \tanh ^{-1}\left (c \sqrt {x}\right )^2\right )\right )}{c^4}}{6 e^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {artanh}\left (c \sqrt {x}\right ) + a x^{2}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 651, normalized size = 1.42 \[ -\frac {a d x}{e^{2}}+\frac {a \,x^{2}}{2 e}+\frac {a \,d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right ) x d}{e^{2}}+\frac {b \arctanh \left (c \sqrt {x}\right ) x^{2}}{2 e}+\frac {b \arctanh \left (c \sqrt {x}\right ) d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}+\frac {b \,x^{\frac {3}{2}}}{6 c e}-\frac {b d \sqrt {x}}{c \,e^{2}}+\frac {b \sqrt {x}}{2 c^{3} e}-\frac {b \ln \left (c \sqrt {x}-1\right ) d}{2 c^{2} e^{2}}+\frac {b \ln \left (c \sqrt {x}-1\right )}{4 c^{4} e}+\frac {b \ln \left (1+c \sqrt {x}\right ) d}{2 c^{2} e^{2}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{4 c^{4} e}+\frac {b \,d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {b \,d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}-\frac {b \,d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}-\frac {b \,d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}+\frac {b \,d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}+\frac {b \,d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}} \]
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 \[ \frac {1}{2} \, a {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + b \int \frac {x^{2} \log \left (c \sqrt {x} + 1\right )}{2 \, {\left (e x + d\right )}}\,{d x} - b \int \frac {x^{2} \log \left (-c \sqrt {x} + 1\right )}{2 \, {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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