Optimal. Leaf size=149 \[ -\frac{\sqrt{c+d x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 c^2 x^2}-\frac{d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{5/2}}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a \left (c+d x^2\right )^{3/2} (4 b c-a d)}{8 c^2 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.148849, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 89, 78, 47, 63, 208} \[ -\frac{\sqrt{c+d x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 c^2 x^2}-\frac{d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{5/2}}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a \left (c+d x^2\right )^{3/2} (4 b c-a d)}{8 c^2 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 89
Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 \sqrt{c+d x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{3}{2} a (4 b c-a d)+3 b^2 c x\right ) \sqrt{c+d x}}{x^3} \, dx,x,x^2\right )}{6 c}\\ &=-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a (4 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^4}+\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^2} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \sqrt{c+d x^2}}{16 c^2 x^2}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a (4 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^4}+\frac{\left (d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{32 c^2}\\ &=-\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \sqrt{c+d x^2}}{16 c^2 x^2}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a (4 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^4}+\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{16 c^2}\\ &=-\frac{\left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \sqrt{c+d x^2}}{16 c^2 x^2}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a (4 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^4}-\frac{d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.117862, size = 142, normalized size = 0.95 \[ \frac{-\left (c+d x^2\right ) \left (a^2 \left (8 c^2+2 c d x^2-3 d^2 x^4\right )+12 a b c x^2 \left (2 c+d x^2\right )+24 b^2 c^2 x^4\right )-3 d x^6 \sqrt{\frac{d x^2}{c}+1} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\sqrt{\frac{d x^2}{c}+1}\right )}{48 c^2 x^6 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 281, normalized size = 1.9 \begin{align*} -{\frac{{a}^{2}}{6\,c{x}^{6}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}d}{8\,{c}^{2}{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}}{16\,{c}^{3}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}+{\frac{{a}^{2}{d}^{3}}{16\,{c}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{ab}{2\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{abd}{4\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{ab{d}^{2}}{4\,{c}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{{b}^{2}d}{2\,c}\sqrt{d{x}^{2}+c}} \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 [A] time = 1.74616, size = 608, normalized size = 4.08 \begin{align*} \left [\frac{3 \,{\left (8 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt{c} x^{6} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \,{\left (8 \, a^{2} c^{3} + 3 \,{\left (8 \, b^{2} c^{3} + 4 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} + a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{96 \, c^{3} x^{6}}, \frac{3 \,{\left (8 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt{-c} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (8 \, a^{2} c^{3} + 3 \,{\left (8 \, b^{2} c^{3} + 4 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} + a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{48 \, c^{3} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 88.9726, size = 291, normalized size = 1.95 \begin{align*} - \frac{a^{2} c}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 a^{2} \sqrt{d}}{24 x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{3}{2}}}{48 c x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{5}{2}}}{16 c^{2} x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 c^{\frac{5}{2}}} - \frac{a b c}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b \sqrt{d}}{4 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a b d^{\frac{3}{2}}}{4 c x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4 c^{\frac{3}{2}}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15071, size = 300, normalized size = 2.01 \begin{align*} \frac{\frac{3 \,{\left (8 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} + 12 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} - 12 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} - 3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} + 8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} + 3 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{c^{2} d^{3} x^{6}}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]