Optimal. Leaf size=406 \[ -\frac {2 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{5/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {f+g x} (e f-d g)^2}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (f+g x)^{3/2} (e f-d g)}-\frac {2 b e^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{5/2}}-\frac {4 b e n}{3 \sqrt {f+g x} (e f-d g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.39, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 51} \[ -\frac {2 b e^{3/2} n \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}-\frac {2 e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{5/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {f+g x} (e f-d g)^2}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (f+g x)^{3/2} (e f-d g)}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{5/2}}-\frac {4 b e n}{3 \sqrt {f+g x} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 51
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2347
Rule 2348
Rule 2402
Rule 2411
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{e f-d g}-\frac {g \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{e (e f-d g)}\\ &=\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {e \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 f-d g)^2}-\frac {g \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{(e f-d g)^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{3 (e f-d g)}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}-\frac {(2 b e n) \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 f-d g)^2}-\frac {(b e n) \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 f-d g)^2}-\frac {(2 b e n) \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 f-d g)^2}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}+\frac {\left (2 b e^{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 f-d g)^{5/2}}-\frac {\left (4 b e^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 g (e f-d g)^2}-\frac {\left (4 b e^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 )}{g (e f-d g)^2}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}+\frac {\left (4 b e^{5/2} 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 f-d g)^{5/2}}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}+\frac {\left (4 b e^{5/2} 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 f-d g)^{5/2}}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}-\frac {\left (4 b e^2 n\right ) \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 f-d g)^3}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}-\frac {4 b e^{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 f-d g)^{5/2}}+\frac {\left (4 b e^2 n\right ) \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 f-d g)^3}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}-\frac {4 b e^{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 f-d g)^{5/2}}-\frac {\left (4 b e^{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 f-d g)^{5/2}}\\ &=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{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 f-d g)^{5/2}}-\frac {4 b e^{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 f-d g)^{5/2}}-\frac {2 b e^{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 f-d g)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.81, size = 608, normalized size = 1.50 \[ \frac {6 e^{3/2} (f+g x)^{3/2} \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-6 e^{3/2} (f+g x)^{3/2} \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+4 (e f-d g)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+12 e (f+g x) \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )-3 b e^{3/2} n (f+g x)^{3/2} \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )+\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )\right )\right )+3 b e^{3/2} n (f+g x)^{3/2} \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )+\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )\right )+24 b e^{3/2} n (f+g x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-8 b e n (f+g x) \sqrt {e f-d g} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )}{6 (f+g x)^{3/2} (e f-d g)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {g x + f} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt {g x + f} a}{e g^{3} x^{4} + d f^{3} + {\left (3 \, e f g^{2} + d g^{3}\right )} x^{3} + 3 \, {\left (e f^{2} g + d f g^{2}\right )} x^{2} + {\left (e f^{3} + 3 \, d f^{2} g\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e x +d \right )^{n}\right )+a}{\left (e x +d \right ) \left (g x +f \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________