Integrand size = 20, antiderivative size = 257 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {\sqrt {-a} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e \left (\sqrt {-a} d-\sqrt {b} e\right ) (1+m) (2+m)}+\frac {\sqrt {-a} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e \left (\sqrt {-a} d+\sqrt {b} e\right ) (1+m) (2+m)}-\frac {2 p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)} \]
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Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2513, 1584, 975, 67, 845, 70} \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}+\frac {\sqrt {-a} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} d-\sqrt {b} e\right )}+\frac {\sqrt {-a} p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} d+\sqrt {b} e\right )}-\frac {2 p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {e x}{d}+1\right )}{d e \left (m^2+3 m+2\right )} \]
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Rule 67
Rule 70
Rule 845
Rule 975
Rule 1584
Rule 2513
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \frac {(d+e x)^{1+m}}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \frac {(d+e x)^{1+m}}{x \left (b+a x^2\right )} \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \left (\frac {(d+e x)^{1+m}}{b x}-\frac {a x (d+e x)^{1+m}}{b \left (b+a x^2\right )}\right ) \, dx}{e (1+m)} \\ & = \frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 p) \int \frac {(d+e x)^{1+m}}{x} \, dx}{e (1+m)}-\frac {(2 a p) \int \frac {x (d+e x)^{1+m}}{b+a x^2} \, dx}{e (1+m)} \\ & = -\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}-\frac {(2 a p) \int \left (-\frac {\sqrt {-a} (d+e x)^{1+m}}{2 a \left (\sqrt {b}-\sqrt {-a} x\right )}+\frac {\sqrt {-a} (d+e x)^{1+m}}{2 a \left (\sqrt {b}+\sqrt {-a} x\right )}\right ) \, dx}{e (1+m)} \\ & = -\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {\left (\sqrt {-a} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {b}-\sqrt {-a} x} \, dx}{e (1+m)}-\frac {\left (\sqrt {-a} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {b}+\sqrt {-a} x} \, dx}{e (1+m)} \\ & = \frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e \left (\sqrt {-a} d-\sqrt {b} e\right ) (1+m) (2+m)}+\frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e \left (\sqrt {-a} d+\sqrt {b} e\right ) (1+m) (2+m)}-\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.82 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {p (d+e x) \left (d \left (a d-\sqrt {-a} \sqrt {b} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+d \left (a d+\sqrt {-a} \sqrt {b} e\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-2 \left (a d^2+b e^2\right ) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )\right )}{d \left (a d^2+b e^2\right ) (2+m)}+\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\right )}{e (1+m)} \]
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\[\int \left (e x +d \right )^{m} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )d x\]
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\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\text {Timed out} \]
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\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]
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\[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
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