Optimal. Leaf size=84 \[ -\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}-\frac {2 a \sqrt {c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{\sqrt {d}} \]
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Rubi [A] time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 451, 217, 206} \[ -\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}-\frac {2 a \sqrt {c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{\sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 451
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^4 \sqrt {c+d x^2}} \, dx &=-\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {2 a (3 b c-a d)+3 b^2 c x^2}{x^2 \sqrt {c+d x^2}} \, dx}{3 c}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}-\frac {2 a (3 b c-a d) \sqrt {c+d x^2}}{3 c^2 x}+b^2 \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=-\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}-\frac {2 a (3 b c-a d) \sqrt {c+d x^2}}{3 c^2 x}+b^2 \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=-\frac {a^2 \sqrt {c+d x^2}}{3 c x^3}-\frac {2 a (3 b c-a d) \sqrt {c+d x^2}}{3 c^2 x}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{\sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 0.86 \[ \frac {b^2 \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{\sqrt {d}}-\frac {a \sqrt {c+d x^2} \left (a \left (c-2 d x^2\right )+6 b c x^2\right )}{3 c^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 173, normalized size = 2.06 \[ \left [\frac {3 \, b^{2} c^{2} \sqrt {d} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (a^{2} c d + 2 \, {\left (3 \, a b c d - a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, c^{2} d x^{3}}, -\frac {3 \, b^{2} c^{2} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a^{2} c d + 2 \, {\left (3 \, a b c d - a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3 \, c^{2} d x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 156, normalized size = 1.86 \[ -\frac {b^{2} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, \sqrt {d}} + \frac {4 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b \sqrt {d} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} + 3 \, a b c^{2} \sqrt {d} - a^{2} c d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 1.01 \[ \frac {b^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{\sqrt {d}}+\frac {2 \sqrt {d \,x^{2}+c}\, a^{2} d}{3 c^{2} x}-\frac {2 \sqrt {d \,x^{2}+c}\, a b}{c x}-\frac {\sqrt {d \,x^{2}+c}\, a^{2}}{3 c \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 77, normalized size = 0.92 \[ \frac {b^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {d}} - \frac {2 \, \sqrt {d x^{2} + c} a b}{c x} + \frac {2 \, \sqrt {d x^{2} + c} a^{2} d}{3 \, c^{2} x} - \frac {\sqrt {d x^{2} + c} a^{2}}{3 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{x^4\,\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.42, size = 158, normalized size = 1.88 \[ - \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 c x^{2}} + \frac {2 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3 c^{2}} - \frac {2 a b \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{c} + b^{2} \left (\begin {cases} \frac {\sqrt {- \frac {c}{d}} \operatorname {asin}{\left (x \sqrt {- \frac {d}{c}} \right )}}{\sqrt {c}} & \text {for}\: c > 0 \wedge d < 0 \\\frac {\sqrt {\frac {c}{d}} \operatorname {asinh}{\left (x \sqrt {\frac {d}{c}} \right )}}{\sqrt {c}} & \text {for}\: c > 0 \wedge d > 0 \\\frac {\sqrt {- \frac {c}{d}} \operatorname {acosh}{\left (x \sqrt {- \frac {d}{c}} \right )}}{\sqrt {- c}} & \text {for}\: d > 0 \wedge c < 0 \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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