Integrand size = 20, antiderivative size = 205 \[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {\sqrt {b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e \left (\sqrt {b} d-\sqrt {-a} e\right ) (1+m) (2+m)}+\frac {\sqrt {b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e \left (\sqrt {b} d+\sqrt {-a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)} \]
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Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 845, 70} \[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c \left (a+b x^2\right )^p\right )}{e (m+1)}+\frac {\sqrt {b} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {b} d-\sqrt {-a} e\right )}+\frac {\sqrt {b} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} e+\sqrt {b} d\right )} \]
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Rule 70
Rule 845
Rule 2513
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.86 \[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {\sqrt {b} p (d+e x) \left (\left (\sqrt {b} d+\sqrt {-a} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+\left (\sqrt {b} d-\sqrt {-a} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )\right )}{\left (b d^2+a e^2\right ) (2+m)}+\log \left (c \left (a+b x^2\right )^p\right )\right )}{e (1+m)} \]
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\[\int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )d x\]
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\[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\text {Timed out} \]
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\[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \,d x } \]
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\[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\int \ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
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