Optimal. Leaf size=185 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{6 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^6} (2 b c-a d)}{6 a^2 c \left (a+b x^6\right ) (b c-a d)}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.227679, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 103, 151, 156, 63, 208} \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{6 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^6} (2 b c-a d)}{6 a^2 c \left (a+b x^6\right ) (b c-a d)}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 103
Rule 151
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2 \sqrt{c+d x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (4 b c+a d)+\frac{3 b d x}{2}}{x (a+b x)^2 \sqrt{c+d x}} \, dx,x,x^6\right )}{6 a c}\\ &=-\frac{b (2 b c-a d) \sqrt{c+d x^6}}{6 a^2 c (b c-a d) \left (a+b x^6\right )}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (b c-a d) (4 b c+a d)+\frac{1}{2} b d (2 b c-a d) x}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^6\right )}{6 a^2 c (b c-a d)}\\ &=-\frac{b (2 b c-a d) \sqrt{c+d x^6}}{6 a^2 c (b c-a d) \left (a+b x^6\right )}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}+\frac{\left (b^2 (4 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^6\right )}{12 a^3 (b c-a d)}-\frac{(4 b c+a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^6\right )}{12 a^3 c}\\ &=-\frac{b (2 b c-a d) \sqrt{c+d x^6}}{6 a^2 c (b c-a d) \left (a+b x^6\right )}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}+\frac{\left (b^2 (4 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^6}\right )}{6 a^3 d (b c-a d)}-\frac{(4 b c+a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^6}\right )}{6 a^3 c d}\\ &=-\frac{b (2 b c-a d) \sqrt{c+d x^6}}{6 a^2 c (b c-a d) \left (a+b x^6\right )}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}+\frac{(4 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{6 a^3 c^{3/2}}-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 a^3 (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.537189, size = 163, normalized size = 0.88 \[ \frac{\frac{a \sqrt{c+d x^6} \left (a^2 d+a b \left (d x^6-c\right )-2 b^2 c x^6\right )}{x^6 \left (a+b x^6\right ) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{\sqrt{c}}}{6 a^3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, 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{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.36195, size = 2525, normalized size = 13.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A] time = 1.19547, size = 362, normalized size = 1.96 \begin{align*} \frac{1}{6} \, d^{3}{\left (\frac{{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{6} + c\right )}^{\frac{3}{2}} b^{2} c - 2 \, \sqrt{d x^{6} + c} b^{2} c^{2} -{\left (d x^{6} + c\right )}^{\frac{3}{2}} a b d + 2 \, \sqrt{d x^{6} + c} a b c d - \sqrt{d x^{6} + c} a^{2} d^{2}}{{\left (a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left ({\left (d x^{6} + c\right )}^{2} b - 2 \,{\left (d x^{6} + c\right )} b c + b c^{2} +{\left (d x^{6} + c\right )} a d - a c d\right )}} - \frac{{\left (4 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{6} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c d^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]