Optimal. Leaf size=87 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0654591, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1876, 212, 208, 205, 275} \[ \frac{c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1876
Rule 212
Rule 208
Rule 205
Rule 275
Rubi steps
\begin{align*} \int \frac{c+d x}{a-b x^4} \, dx &=\int \left (\frac{c}{a-b x^4}+\frac{d x}{a-b x^4}\right ) \, dx\\ &=c \int \frac{1}{a-b x^4} \, dx+d \int \frac{x}{a-b x^4} \, dx\\ &=\frac{c \int \frac{1}{\sqrt{a}-\sqrt{b} x^2} \, dx}{2 \sqrt{a}}+\frac{c \int \frac{1}{\sqrt{a}+\sqrt{b} x^2} \, dx}{2 \sqrt{a}}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )\\ &=\frac{c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0353374, size = 134, normalized size = 1.54 \[ \frac{-\left (\sqrt [4]{a} d+\sqrt [4]{b} c\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \sqrt [4]{b} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 101, normalized size = 1.2 \begin{align*}{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{4}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.770489, size = 126, normalized size = 1.45 \begin{align*} - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b d^{2} - 16 t a b c^{2} d + a d^{4} - b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 128 t^{3} a^{3} b d^{2} + 16 t^{2} a^{2} b c^{2} d + 8 t a^{2} d^{4} - 4 t a b c^{4} + 5 a c^{2} d^{3}}{4 a c d^{4} + b c^{5}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.09158, size = 304, normalized size = 3.49 \begin{align*} \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b} - \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b d + \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b d + \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]