Optimal. Leaf size=143 \[ \frac {3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2389, 2296, 2300, 2180, 2204} \[ \frac {3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2296
Rule 2300
Rule 2389
Rubi steps
\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {(3 b n) \operatorname {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 e}\\ &=-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e}\\ &=-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b^2 n (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{4 e}\\ &=-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b n (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{2 e}\\ &=\frac {3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 127, normalized size = 0.89 \[ \frac {(d+e x) \left (3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a+2 b \log \left (c (d+e x)^n\right )-3 b n\right )\right )}{4 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________