Optimal. Leaf size=659 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g+2 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{40 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b (b c-4 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{3 b^{4/3} e \sqrt{a+b x^3}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \sqrt{a+b x^3} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right )-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.979823, antiderivative size = 659, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g+2 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{40 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b (b c-4 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{3 b^{4/3} e \sqrt{a+b x^3}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \sqrt{a+b x^3} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right )-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 1825
Rule 1835
Rule 1832
Rule 266
Rule 63
Rule 208
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{1}{2} (3 b) \int \frac{-\frac{c}{6}-\frac{d x}{5}-\frac{e x^2}{4}-\frac{f x^3}{3}-\frac{g x^4}{2}}{x^4 \sqrt{a+b x^3}} \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}+\frac{b \int \frac{\frac{6 a d}{5}+\frac{3 a e x}{2}-\frac{1}{2} (b c-4 a f) x^2+3 a g x^3}{x^3 \sqrt{a+b x^3}} \, dx}{4 a}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{b \int \frac{-6 a^2 e+2 a (b c-4 a f) x+\frac{6}{5} a (b d-10 a g) x^2}{x^2 \sqrt{a+b x^3}} \, dx}{16 a^2}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}+\frac{b \int \frac{-4 a^2 (b c-4 a f)-\frac{12}{5} a^2 (b d-10 a g) x+6 a^2 b e x^2}{x \sqrt{a+b x^3}} \, dx}{32 a^3}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}+\frac{b \int \frac{-\frac{12}{5} a^2 (b d-10 a g)+6 a^2 b e x}{\sqrt{a+b x^3}} \, dx}{32 a^3}-\frac{(b (b c-4 a f)) \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{8 a}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}+\frac{\left (3 b^{5/3} e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{16 a}-\frac{(b (b c-4 a f)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a}-\frac{\left (3 b \left (2 b d+5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{80 a}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}+\frac{3 b^{4/3} e \sqrt{a+b x^3}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} e \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}} E\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 )}{16 a^{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{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (2 b d+5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g\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 )}{40 a \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{(b c-4 a f) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{12 a}\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \sqrt{a+b x^3}-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}+\frac{3 b^{4/3} e \sqrt{a+b x^3}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{b (b c-4 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} e \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}} E\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 )}{16 a^{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{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (2 b d+5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g\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 )}{40 a \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 = 0.424939, size = 211, normalized size = 0.32 \[ -\frac{\sqrt{a+b x^3} \left (5 x \left (2 x \left (6 a^2 f \left (a \sqrt{\frac{b x^3}{a}+1}+b x^3 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )\right )+9 a^3 g x \, _2F_1\left (-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right )+4 b^2 c x^3 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right )\right )+9 a^3 e \, _2F_1\left (-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right )\right )+36 a^3 d \, _2F_1\left (-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )\right )}{180 a^3 x^5 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.008, size = 1180, normalized size = 1.8 \begin{align*} \text{result too large to display} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.38378, size = 304, normalized size = 0.46 \begin{align*} \frac{\sqrt{a} d \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, - \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} + \frac{\sqrt{a} e \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} + \frac{\sqrt{a} g \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} - \frac{a c}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b} c}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b} f \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{b^{\frac{3}{2}} c}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b f \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} + \frac{b^{2} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} \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*} \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]