Optimal. Leaf size=119 \[ -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {b d p \log \left (a+b x^2\right )}{e \left (a e^2+b d^2\right )}-\frac {2 b d p \log (d+e x)}{e \left (a e^2+b d^2\right )}+\frac {2 \sqrt {a} \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a e^2+b d^2} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2463, 801, 635, 205, 260} \[ -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {b d p \log \left (a+b x^2\right )}{e \left (a e^2+b d^2\right )}-\frac {2 b d p \log (d+e x)}{e \left (a e^2+b d^2\right )}+\frac {2 \sqrt {a} \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a e^2+b d^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 2463
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {(2 b p) \int \frac {x}{(d+e x) \left (a+b x^2\right )} \, dx}{e}\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {(2 b p) \int \left (-\frac {d e}{\left (b d^2+a e^2\right ) (d+e x)}+\frac {a e+b d x}{\left (b d^2+a e^2\right ) \left (a+b x^2\right )}\right ) \, dx}{e}\\ &=-\frac {2 b d p \log (d+e x)}{e \left (b d^2+a e^2\right )}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {(2 b p) \int \frac {a e+b d x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )}\\ &=-\frac {2 b d p \log (d+e x)}{e \left (b d^2+a e^2\right )}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}+\frac {(2 a b p) \int \frac {1}{a+b x^2} \, dx}{b d^2+a e^2}+\frac {\left (2 b^2 d p\right ) \int \frac {x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )}\\ &=\frac {2 \sqrt {a} \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b d^2+a e^2}-\frac {2 b d p \log (d+e x)}{e \left (b d^2+a e^2\right )}+\frac {b d p \log \left (a+b x^2\right )}{e \left (b d^2+a e^2\right )}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 137, normalized size = 1.15 \[ \frac {-b d^2 \log \left (c \left (a+b x^2\right )^p\right )-a e^2 \log \left (c \left (a+b x^2\right )^p\right )+b d^2 p \log \left (a+b x^2\right )+b d e p x \log \left (a+b x^2\right )+2 \sqrt {a} \sqrt {b} e p (d+e x) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-2 b d p (d+e x) \log (d+e x)}{e (d+e x) \left (a e^2+b d^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 261, normalized size = 2.19 \[ \left [\frac {{\left (e^{2} p x + d e p\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (b d e p x - a e^{2} p\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b d e p x + b d^{2} p\right )} \log \left (e x + d\right ) - {\left (b d^{2} + a e^{2}\right )} \log \relax (c)}{b d^{3} e + a d e^{3} + {\left (b d^{2} e^{2} + a e^{4}\right )} x}, \frac {2 \, {\left (e^{2} p x + d e p\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (b d e p x - a e^{2} p\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b d e p x + b d^{2} p\right )} \log \left (e x + d\right ) - {\left (b d^{2} + a e^{2}\right )} \log \relax (c)}{b d^{3} e + a d e^{3} + {\left (b d^{2} e^{2} + a e^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 158, normalized size = 1.33 \[ \frac {b d p \log \left (b x^{2} + a\right )}{b d^{2} e + a e^{3}} + \frac {2 \, a b p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b d^{2} + a e^{2}\right )} \sqrt {a b}} - \frac {2 \, b d p x e \log \left (x e + d\right ) + b d^{2} p \log \left (b x^{2} + a\right ) + 2 \, b d^{2} p \log \left (x e + d\right ) + a p e^{2} \log \left (b x^{2} + a\right ) + b d^{2} \log \relax (c) + a e^{2} \log \relax (c)}{b d^{2} x e^{2} + b d^{3} e + a x e^{4} + a d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.61, size = 1233, normalized size = 10.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 108, normalized size = 0.91 \[ \frac {{\left (\frac {2 \, a e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b d^{2} + a e^{2}\right )} \sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{b d^{2} + a e^{2}} - \frac {2 \, d \log \left (e x + d\right )}{b d^{2} + a e^{2}}\right )} b p}{e} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 337, normalized size = 2.83 \[ \frac {\ln \left (\frac {4\,b^3\,p^2\,x}{e}-\frac {p\,\left (b\,d+e\,\sqrt {-a\,b}\right )\,\left (2\,a\,b^2\,e\,p+2\,b^3\,d\,p\,x-\frac {2\,b^2\,e\,p\,\left (b\,d+e\,\sqrt {-a\,b}\right )\,\left (-b\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{b\,d^2\,e+a\,e^3}\right )}{b\,d^2\,e+a\,e^3}\right )\,\left (b\,d\,p+e\,p\,\sqrt {-a\,b}\right )}{b\,d^2\,e+a\,e^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{e\,\left (d+e\,x\right )}+\frac {\ln \left (\frac {4\,b^3\,p^2\,x}{e}-\frac {p\,\left (b\,d-e\,\sqrt {-a\,b}\right )\,\left (2\,a\,b^2\,e\,p+2\,b^3\,d\,p\,x-\frac {2\,b^2\,e\,p\,\left (b\,d-e\,\sqrt {-a\,b}\right )\,\left (-b\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{b\,d^2\,e+a\,e^3}\right )}{b\,d^2\,e+a\,e^3}\right )\,\left (b\,d\,p-e\,p\,\sqrt {-a\,b}\right )}{b\,d^2\,e+a\,e^3}-\frac {2\,b\,d\,p\,\ln \left (d+e\,x\right )}{b\,d^2\,e+a\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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