Optimal. Leaf size=332 \[ -\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{a+b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.540226, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2141, 218, 2140, 206} \[ -\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{a+b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2141
Rule 218
Rule 2140
Rule 206
Rubi steps
\begin{align*} \int \frac{e+f x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a+b x^3}} \, dx &=\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt{3}\right )^3 a b\right )+6 a b^{4/3} x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a+b x^3}} \, dx}{12 \sqrt{3} a^{4/3} b^{4/3}}-\frac{\left (6 a b^{4/3} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt{3}\right )^3 a b\right ) f\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{b} \left (-28 a b+\left (1-\sqrt{3}\right )^3 a b\right )}\\ &=-\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \operatorname{Subst}\left (\int \frac{1}{1+\left (3-2 \sqrt{3}\right ) a x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{a+b x^3}}\right )}{\sqrt{3} b^{2/3}}\\ &=-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\sqrt{-3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{a+b x^3}}\right )}{\sqrt{3 \left (-3+2 \sqrt{3}\right )} \sqrt{a} b^{2/3}}-\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 1.76671, size = 438, normalized size = 1.32 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (i \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\left (\sqrt{3}-1\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{i \sqrt [4]{3} f \left (\left (\sqrt{3}+(-2-i)\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )}{2 \sqrt{2}}\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{(fx+e) \left ( \sqrt [3]{b}x+\sqrt [3]{a} \left ( 1-\sqrt{3} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{b x^{3} + a}{\left (b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} - 1\right )}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e + f x}{\sqrt{a + b x^{3}} \left (- \sqrt{3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]