Optimal. Leaf size=88 \[ \frac {2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right )}{c^2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2389, 2299, 2181} \[ \frac {2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right )}{c^2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 2299
Rule 2389
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {2 \operatorname {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \sqrt {d+e x}\right )\right )}{c^2 e}\\ &=\frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p}}{c^2 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 88, normalized size = 1.00 \[ \frac {2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right )}{c^2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (\sqrt {e x + d} c\right ) + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (\sqrt {e x + d} c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (\sqrt {e x +d}\, c \right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.23, size = 59, normalized size = 0.67 \[ -\frac {2 \, {\left (b \log \left (\sqrt {e x + d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac {2 \, a}{b}\right )} E_{-p}\left (-\frac {2 \, {\left (b \log \left (\sqrt {e x + d} c\right ) + a\right )}}{b}\right )}{b c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\ln \left (c\,\sqrt {d+e\,x}\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \sqrt {d + e x} \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________