Optimal. Leaf size=114 \[ -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}-\frac {4 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {4 b e n}{3 g \sqrt {f+g x} (e f-d g)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2395, 51, 63, 208} \[ -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}-\frac {4 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {4 b e n}{3 g \sqrt {f+g x} (e f-d g)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 2395
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{5/2}} \, dx &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}+\frac {(2 b e n) \int \frac {1}{(d+e x) (f+g x)^{3/2}} \, dx}{3 g}\\ &=\frac {4 b e n}{3 g (e f-d g) \sqrt {f+g x}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}+\frac {\left (2 b e^2 n\right ) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{3 g (e f-d g)}\\ &=\frac {4 b e n}{3 g (e f-d g) \sqrt {f+g x}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}+\frac {\left (4 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 g^2 (e f-d g)}\\ &=\frac {4 b e n}{3 g (e f-d g) \sqrt {f+g x}}-\frac {4 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 85, normalized size = 0.75 \[ -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^{3/2}}-\frac {4 b e n \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )}{3 g \sqrt {f+g x} (d g-e f)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 425, normalized size = 3.73 \[ \left [-\frac {2 \, {\left ({\left (b e g^{2} n x^{2} + 2 \, b e f g n x + b e f^{2} n\right )} \sqrt {\frac {e}{e f - d g}} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, {\left (e f - d g\right )} \sqrt {g x + f} \sqrt {\frac {e}{e f - d g}}}{e x + d}\right ) - {\left (2 \, b e g n x + 2 \, b e f n - a e f + a d g - {\left (b e f - b d g\right )} n \log \left (e x + d\right ) - {\left (b e f - b d g\right )} \log \relax (c)\right )} \sqrt {g x + f}\right )}}{3 \, {\left (e f^{3} g - d f^{2} g^{2} + {\left (e f g^{3} - d g^{4}\right )} x^{2} + 2 \, {\left (e f^{2} g^{2} - d f g^{3}\right )} x\right )}}, -\frac {2 \, {\left (2 \, {\left (b e g^{2} n x^{2} + 2 \, b e f g n x + b e f^{2} n\right )} \sqrt {-\frac {e}{e f - d g}} \arctan \left (-\frac {{\left (e f - d g\right )} \sqrt {g x + f} \sqrt {-\frac {e}{e f - d g}}}{e g x + e f}\right ) - {\left (2 \, b e g n x + 2 \, b e f n - a e f + a d g - {\left (b e f - b d g\right )} n \log \left (e x + d\right ) - {\left (b e f - b d g\right )} \log \relax (c)\right )} \sqrt {g x + f}\right )}}{3 \, {\left (e f^{3} g - d f^{2} g^{2} + {\left (e f g^{3} - d g^{4}\right )} x^{2} + 2 \, {\left (e f^{2} g^{2} - d f g^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 188, normalized size = 1.65 \[ -\frac {4 \, b n \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right ) e^{2}}{3 \, {\left (d g^{2} - f g e\right )} \sqrt {d g e - f e^{2}}} - \frac {2 \, {\left (b d g n \log \left (d g + {\left (g x + f\right )} e - f e\right ) - b f n e \log \left (d g + {\left (g x + f\right )} e - f e\right ) - b d g n \log \relax (g) + b f n e \log \relax (g) + 2 \, {\left (g x + f\right )} b n e + b d g \log \relax (c) - b f e \log \relax (c) + a d g - a f e\right )}}{3 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} d g^{2} - {\left (g x + f\right )}^{\frac {3}{2}} f g e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e x +d \right )^{n}\right )+a}{\left (g x +f \right )^{\frac {5}{2}}}\, dx \]
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 = -0.01 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 76.04, size = 117, normalized size = 1.03 \[ \frac {- \frac {2 a}{3 \left (f + g x\right )^{\frac {3}{2}}} + 2 b \left (\frac {2 e n \left (- \frac {g}{\sqrt {f + g x} \left (d g - e f\right )} - \frac {g \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{\sqrt {\frac {d g - e f}{e}} \left (d g - e f\right )}\right )}{3 g} - \frac {\log {\left (c \left (d - \frac {e f}{g} + \frac {e \left (f + g x\right )}{g}\right )^{n} \right )}}{3 \left (f + g x\right )^{\frac {3}{2}}}\right )}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________