Optimal. Leaf size=205 \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^2\right )^p\right )}{e (m+1)}+\frac {\sqrt {b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {b} d-\sqrt {-a} e\right )}+\frac {\sqrt {b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} e+\sqrt {b} d\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2463, 831, 68} \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^2\right )^p\right )}{e (m+1)}+\frac {\sqrt {b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {b} d-\sqrt {-a} e\right )}+\frac {\sqrt {b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} e+\sqrt {b} d\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 831
Rule 2463
Rubi steps
\begin {align*} \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}-\frac {(2 b p) \int \frac {x (d+e x)^{1+m}}{a+b x^2} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}-\frac {(2 b p) \int \left (-\frac {(d+e x)^{1+m}}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {(d+e x)^{1+m}}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}+\frac {\left (\sqrt {b} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {-a}-\sqrt {b} x} \, dx}{e (1+m)}-\frac {\left (\sqrt {b} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {-a}+\sqrt {b} x} \, dx}{e (1+m)}\\ &=\frac {\sqrt {b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e \left (\sqrt {b} d-\sqrt {-a} e\right ) (1+m) (2+m)}+\frac {\sqrt {b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e \left (\sqrt {b} d+\sqrt {-a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 176, normalized size = 0.86 \[ \frac {(d+e x)^{m+1} \left (\log \left (c \left (a+b x^2\right )^p\right )+\frac {\sqrt {b} p (d+e x) \left (\left (\sqrt {-a} e+\sqrt {b} d\right ) \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+\left (\sqrt {b} d-\sqrt {-a} e\right ) \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )\right )}{(m+2) \left (a e^2+b d^2\right )}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )}^{m} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{m} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.36, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e p x + d p\right )} {\left (e x + d\right )}^{m} \log \left (b x^{2} + a\right )}{e {\left (m + 1\right )}} + \int -\frac {{\left (2 \, b d p x - {\left (e {\left (m + 1\right )} \log \relax (c) - 2 \, e p\right )} b x^{2} - a e {\left (m + 1\right )} \log \relax (c)\right )} {\left (e x + d\right )}^{m}}{b e {\left (m + 1\right )} x^{2} + a e {\left (m + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
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]
________________________________________________________________________________________