Optimal. Leaf size=174 \[ \frac {b d p}{e \left (b d^2+a e^2\right ) (d+e x)}+\frac {2 \sqrt {a} b^{3/2} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\left (b d^2+a e^2\right )^2}-\frac {b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}+\frac {b \left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 e \left (b d^2+a e^2\right )^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 174, 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 = {2513, 815, 649,
211, 266} \begin {gather*} \frac {2 \sqrt {a} b^{3/2} d p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\left (a e^2+b d^2\right )^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac {b p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 e \left (a e^2+b d^2\right )^2}+\frac {b d p}{e (d+e x) \left (a e^2+b d^2\right )}-\frac {b p \left (b d^2-a e^2\right ) \log (d+e x)}{e \left (a e^2+b d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 2513
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^3} \, dx &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac {(b p) \int \frac {x}{(d+e x)^2 \left (a+b x^2\right )} \, dx}{e}\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac {(b p) \int \left (-\frac {d e}{\left (b d^2+a e^2\right ) (d+e x)^2}+\frac {e \left (-b d^2+a e^2\right )}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac {b \left (2 a d e+\left (b d^2-a e^2\right ) x\right )}{\left (b d^2+a e^2\right )^2 \left (a+b x^2\right )}\right ) \, dx}{e}\\ &=\frac {b d p}{e \left (b d^2+a e^2\right ) (d+e x)}-\frac {b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac {\left (b^2 p\right ) \int \frac {2 a d e+\left (b d^2-a e^2\right ) x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac {b d p}{e \left (b d^2+a e^2\right ) (d+e x)}-\frac {b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac {\left (2 a b^2 d p\right ) \int \frac {1}{a+b x^2} \, dx}{\left (b d^2+a e^2\right )^2}+\frac {\left (b^2 \left (b d^2-a e^2\right ) p\right ) \int \frac {x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac {b d p}{e \left (b d^2+a e^2\right ) (d+e x)}+\frac {2 \sqrt {a} b^{3/2} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\left (b d^2+a e^2\right )^2}-\frac {b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}+\frac {b \left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 e \left (b d^2+a e^2\right )^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 217, normalized size = 1.25 \begin {gather*} \frac {\frac {b p (d+e x) \left (\left (\sqrt {-a} b d^2+2 a \sqrt {b} d e+(-a)^{3/2} e^2\right ) (d+e x) \log \left (\sqrt {-a}-\sqrt {b} x\right )+\left (\sqrt {-a} b d^2-2 a \sqrt {b} d e+(-a)^{3/2} e^2\right ) (d+e x) \log \left (\sqrt {-a}+\sqrt {b} x\right )+2 \sqrt {-a} \left (b d^3+a d e^2-\left (b d^2-a e^2\right ) (d+e x) \log (d+e x)\right )\right )}{\sqrt {-a} \left (b d^2+a e^2\right )^2}-\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.90, size = 2684, normalized size = 15.43
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2684\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 198, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, {\left (\frac {4 \, a b d \arctan \left (\frac {b x}{\sqrt {a b}}\right ) e}{{\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a b}} + \frac {{\left (b d^{2} - a e^{2}\right )} \log \left (b x^{2} + a\right )}{b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}} - \frac {2 \, {\left (b d^{2} - a e^{2}\right )} \log \left (x e + d\right )}{b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}} + \frac {2 \, d}{b d^{3} + a d e^{2} + {\left (b d^{2} e + a e^{3}\right )} x}\right )} b p e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (155) = 310\).
time = 0.42, size = 724, normalized size = 4.16 \begin {gather*} \left [\frac {2 \, b^{2} d^{3} p x e + 2 \, b^{2} d^{4} p + 2 \, a b d p x e^{3} + 2 \, a b d^{2} p e^{2} + 2 \, {\left (b d p x^{2} e^{3} + 2 \, b d^{2} p x e^{2} + b d^{3} p e\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (2 \, b^{2} d^{3} p x e - 2 \, a b d p x e^{3} - {\left (a b p x^{2} + a^{2} p\right )} e^{4} + {\left (b^{2} d^{2} p x^{2} - 3 \, a b d^{2} p\right )} e^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (2 \, b^{2} d^{3} p x e + b^{2} d^{4} p - a b p x^{2} e^{4} - 2 \, a b d p x e^{3} + {\left (b^{2} d^{2} p x^{2} - a b d^{2} p\right )} e^{2}\right )} \log \left (x e + d\right ) - {\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c\right )}{2 \, {\left (2 \, b^{2} d^{5} x e^{2} + b^{2} d^{6} e + 4 \, a b d^{3} x e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + {\left (2 \, a b d^{2} x^{2} + a^{2} d^{2}\right )} e^{5} + {\left (b^{2} d^{4} x^{2} + 2 \, a b d^{4}\right )} e^{3}\right )}}, \frac {2 \, b^{2} d^{3} p x e + 2 \, b^{2} d^{4} p + 2 \, a b d p x e^{3} + 2 \, a b d^{2} p e^{2} + 4 \, {\left (b d p x^{2} e^{3} + 2 \, b d^{2} p x e^{2} + b d^{3} p e\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (2 \, b^{2} d^{3} p x e - 2 \, a b d p x e^{3} - {\left (a b p x^{2} + a^{2} p\right )} e^{4} + {\left (b^{2} d^{2} p x^{2} - 3 \, a b d^{2} p\right )} e^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (2 \, b^{2} d^{3} p x e + b^{2} d^{4} p - a b p x^{2} e^{4} - 2 \, a b d p x e^{3} + {\left (b^{2} d^{2} p x^{2} - a b d^{2} p\right )} e^{2}\right )} \log \left (x e + d\right ) - {\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c\right )}{2 \, {\left (2 \, b^{2} d^{5} x e^{2} + b^{2} d^{6} e + 4 \, a b d^{3} x e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + {\left (2 \, a b d^{2} x^{2} + a^{2} d^{2}\right )} e^{5} + {\left (b^{2} d^{4} x^{2} + 2 \, a b d^{4}\right )} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 420 vs.
\(2 (155) = 310\).
time = 4.22, size = 420, normalized size = 2.41 \begin {gather*} \frac {2 \, a b^{2} d p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a b}} + \frac {{\left (b^{2} d^{2} p - a b p e^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} d^{4} e + 2 \, a b d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac {2 \, b^{2} d^{2} p x^{2} e^{2} \log \left (x e + d\right ) + 4 \, b^{2} d^{3} p x e \log \left (x e + d\right ) - 2 \, b^{2} d^{3} p x e + b^{2} d^{4} p \log \left (b x^{2} + a\right ) + 2 \, b^{2} d^{4} p \log \left (x e + d\right ) - 2 \, b^{2} d^{4} p + 2 \, a b d^{2} p e^{2} \log \left (b x^{2} + a\right ) - 2 \, a b p x^{2} e^{4} \log \left (x e + d\right ) - 4 \, a b d p x e^{3} \log \left (x e + d\right ) - 2 \, a b d^{2} p e^{2} \log \left (x e + d\right ) + b^{2} d^{4} \log \left (c\right ) - 2 \, a b d p x e^{3} - 2 \, a b d^{2} p e^{2} + 2 \, a b d^{2} e^{2} \log \left (c\right ) + a^{2} p e^{4} \log \left (b x^{2} + a\right ) + a^{2} e^{4} \log \left (c\right )}{2 \, {\left (b^{2} d^{4} x^{2} e^{3} + 2 \, b^{2} d^{5} x e^{2} + b^{2} d^{6} e + 2 \, a b d^{2} x^{2} e^{5} + 4 \, a b d^{3} x e^{4} + 2 \, a b d^{4} e^{3} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + a^{2} d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 272, normalized size = 1.56 \begin {gather*} \frac {\ln \left (b^2\,x+\sqrt {-a\,b^3}\right )\,\left (b^2\,d^2\,p-a\,b\,e^2\,p+2\,d\,e\,p\,\sqrt {-a\,b^3}\right )}{2\,\left (a^2\,e^5+2\,a\,b\,d^2\,e^3+b^2\,d^4\,e\right )}-\frac {\ln \left (d+e\,x\right )\,\left (b^2\,d^2\,p-a\,b\,e^2\,p\right )}{a^2\,e^5+2\,a\,b\,d^2\,e^3+b^2\,d^4\,e}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}-\frac {\ln \left (b^2\,x-\sqrt {-a\,b^3}\right )\,\left (a\,b\,e^2\,p-b^2\,d^2\,p+2\,d\,e\,p\,\sqrt {-a\,b^3}\right )}{2\,\left (a^2\,e^5+2\,a\,b\,d^2\,e^3+b^2\,d^4\,e\right )}+\frac {b\,d\,p}{\left (x\,e^2+d\,e\right )\,\left (b\,d^2+a\,e^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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