Optimal. Leaf size=42 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0573733, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1586, 635, 205, 260} \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx &=\int \frac{a c+a d x+b c x^2+b d x^3}{\left (a+b x^2\right )^2} \, dx\\ &=\int \frac{c+d x}{a+b x^2} \, dx\\ &=c \int \frac{1}{a+b x^2} \, dx+d \int \frac{x}{a+b x^2} \, dx\\ &=\frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0143338, size = 42, normalized size = 1. \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{d \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.8 \begin{align*}{\frac{d\ln \left ( b{x}^{2}+a \right ) }{2\,b}}+{c\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33557, size = 225, normalized size = 5.36 \begin{align*} \left [\frac{a d \log \left (b x^{2} + a\right ) - \sqrt{-a b} c \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac{a d \log \left (b x^{2} + a\right ) + 2 \, \sqrt{a b} c \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \, a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.287324, size = 124, normalized size = 2.95 \begin{align*} \left (\frac{d}{2 b} - \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) \log{\left (x + \frac{2 a b \left (\frac{d}{2 b} - \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} + \left (\frac{d}{2 b} + \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) \log{\left (x + \frac{2 a b \left (\frac{d}{2 b} + \frac{c \sqrt{- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22604, size = 42, normalized size = 1. \begin{align*} \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} + \frac{d \log \left (b x^{2} + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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