Optimal. Leaf size=417 \[ -\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}+\frac {2 \sqrt {f+g x} (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 b n (e f-d g)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}+\frac {2 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {16 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}-\frac {4 b n (e f-d g)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2}}-\frac {16 b n \sqrt {f+g x} (e f-d g)}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e} \]
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Rubi [A] time = 1.38, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2411, 2346, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 50} \[ -\frac {2 b n (e f-d g)^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}+\frac {2 \sqrt {f+g x} (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {16 b n \sqrt {f+g x} (e f-d g)}{3 e^2}+\frac {2 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {16 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}-\frac {4 b n (e f-d g)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2}}-\frac {4 b n (f+g x)^{3/2}}{9 e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2346
Rule 2348
Rule 2402
Rule 2411
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=\frac {g \operatorname {Subst}\left (\int \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac {(e f-d g) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}+\frac {(g (e f-d g)) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^3}+\frac {(e f-d g)^2 \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^3}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}}{x} \, dx,x,d+e x\right )}{3 e}\\ &=-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}-\frac {(2 b (e f-d g) n) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{3 e^2}-\frac {(2 b (e f-d g) n) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{e^2}-\frac {\left (b (e f-d g)^2 n\right ) \operatorname {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{e^3}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}+\frac {\left (2 b (e f-d g)^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{e^{5/2}}-\frac {\left (2 b (e f-d g)^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 e^3}-\frac {\left (2 b (e f-d g)^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^3}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}+\frac {\left (4 b (e f-d g)^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{e^{3/2}}-\frac {\left (4 b (e f-d g)^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 e^2 g}-\frac {\left (4 b (e f-d g)^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e^2 g}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {16 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}+\frac {\left (4 b (e f-d g)^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{e^{3/2}}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {16 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}+\frac {2 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}-\frac {(4 b (e f-d g) n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{e^2}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {16 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}+\frac {2 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}-\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}+\frac {(4 b (e f-d g) n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{e^2}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {16 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}+\frac {2 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}-\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}-\frac {\left (4 b (e f-d g)^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}\\ &=-\frac {16 b (e f-d g) n \sqrt {f+g x}}{3 e^2}-\frac {4 b n (f+g x)^{3/2}}{9 e}+\frac {16 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{5/2}}+\frac {2 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{5/2}}+\frac {2 (e f-d g) \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{5/2}}-\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}-\frac {2 b (e f-d g)^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.14, size = 607, normalized size = 1.46 \[ \frac {12 e^{3/2} (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+18 (e f-d g)^{3/2} \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-18 (e f-d g)^{3/2} \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+36 a \sqrt {e} \sqrt {f+g x} (e f-d g)+36 b \sqrt {e} \sqrt {f+g x} (e f-d g) \log \left (c (d+e x)^n\right )-9 b n (e f-d g)^{3/2} \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )+\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )\right )\right )+9 b n (e f-d g)^{3/2} \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )+\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )\right )-96 b n (e f-d g) \left (\sqrt {e} \sqrt {f+g x}-\sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )-8 b e^{3/2} n (f+g x)^{3/2}}{18 e^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x + b f\right )} \sqrt {g x + f} \log \left ({\left (e x + d\right )}^{n} c\right ) + {\left (a g x + a f\right )} \sqrt {g x + f}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{\frac {3}{2}} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{\frac {3}{2}} \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{d+e\,x} \,d x \]
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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