Optimal. Leaf size=143 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}+\frac {\sqrt {c+d x^2} \left (a d (8 b c-a d)+8 b^2 c^2\right )}{8 c^2}-\frac {\left (a d (8 b c-a d)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{8 c^{3/2}}-\frac {a \left (c+d x^2\right )^{3/2} (8 b c-a d)}{8 c^2 x^2} \]
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Rubi [A] time = 0.16, antiderivative size = 140, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 89, 78, 50, 63, 208} \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}+\frac {1}{8} \sqrt {c+d x^2} \left (\frac {a d (8 b c-a d)}{c^2}+8 b^2\right )-\frac {\left (a d (8 b c-a d)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{8 c^{3/2}}-\frac {a \left (c+d x^2\right )^{3/2} (8 b c-a d)}{8 c^2 x^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 \sqrt {c+d x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}+\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {1}{2} a (8 b c-a d)+2 b^2 c x\right ) \sqrt {c+d x}}{x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}-\frac {a (8 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^2}+\frac {1}{16} \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{8} \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}-\frac {a (8 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^2}+\frac {1}{16} \left (c \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {1}{8} \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}-\frac {a (8 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^2}+\frac {\left (c \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{8 d}\\ &=\frac {1}{8} \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}-\frac {a (8 b c-a d) \left (c+d x^2\right )^{3/2}}{8 c^2 x^2}-\frac {1}{8} \sqrt {c} \left (8 b^2+\frac {a d (8 b c-a d)}{c^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 104, normalized size = 0.73 \[ \frac {\sqrt {c+d x^2} \left (-a^2 \left (2 c+d x^2\right )-8 a b c x^2+8 b^2 c x^4\right )}{8 c x^4}-\frac {\left (-a^2 d^2+8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 225, normalized size = 1.57 \[ \left [-\frac {{\left (8 \, b^{2} c^{2} + 8 \, a b c d - a^{2} d^{2}\right )} \sqrt {c} x^{4} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (8 \, b^{2} c^{2} x^{4} - 2 \, a^{2} c^{2} - {\left (8 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{16 \, c^{2} x^{4}}, \frac {{\left (8 \, b^{2} c^{2} + 8 \, a b c d - a^{2} d^{2}\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (8 \, b^{2} c^{2} x^{4} - 2 \, a^{2} c^{2} - {\left (8 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, c^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 153, normalized size = 1.07 \[ \frac {8 \, \sqrt {d x^{2} + c} b^{2} d + \frac {{\left (8 \, b^{2} c^{2} d + 8 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c d^{2} - 8 \, \sqrt {d x^{2} + c} a b c^{2} d^{2} + {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3} + \sqrt {d x^{2} + c} a^{2} c d^{3}}{c d^{2} x^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 207, normalized size = 1.45 \[ \frac {a^{2} d^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{8 c^{\frac {3}{2}}}-\frac {a b d \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{\sqrt {c}}-b^{2} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )-\frac {\sqrt {d \,x^{2}+c}\, a^{2} d^{2}}{8 c^{2}}+\frac {\sqrt {d \,x^{2}+c}\, a b d}{c}+\sqrt {d \,x^{2}+c}\, b^{2}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} d}{8 c^{2} x^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b}{c \,x^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2}}{4 c \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 173, normalized size = 1.21 \[ -b^{2} \sqrt {c} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {a b d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{\sqrt {c}} + \frac {a^{2} d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{8 \, c^{\frac {3}{2}}} + \sqrt {d x^{2} + c} b^{2} + \frac {\sqrt {d x^{2} + c} a b d}{c} - \frac {\sqrt {d x^{2} + c} a^{2} d^{2}}{8 \, c^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{c x^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{8 \, c^{2} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{4 \, c x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 137, normalized size = 0.96 \[ b^2\,\sqrt {d\,x^2+c}-\frac {\left (\frac {a^2\,d^2}{8}-a\,b\,c\,d\right )\,\sqrt {d\,x^2+c}+\frac {\left (a^2\,d^2+8\,b\,c\,a\,d\right )\,{\left (d\,x^2+c\right )}^{3/2}}{8\,c}}{{\left (d\,x^2+c\right )}^2-2\,c\,\left (d\,x^2+c\right )+c^2}-\frac {\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (-a^2\,d^2+8\,a\,b\,c\,d+8\,b^2\,c^2\right )}{8\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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