Optimal. Leaf size=111 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {b x \sqrt {c+d x^2} (4 a d+b c)}{2 c}+\frac {b (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d}} \]
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Rubi [A] time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {462, 453, 195, 217, 206} \begin {gather*} -\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {b x \sqrt {c+d x^2} (4 a d+b c)}{2 c}+\frac {b (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 453
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^4} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}+\frac {\int \frac {\left (6 a b c+3 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^2} \, dx}{3 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {(b (b c+4 a d)) \int \sqrt {c+d x^2} \, dx}{c}\\ &=\frac {b (b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {1}{2} (b (b c+4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=\frac {b (b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {1}{2} (b (b c+4 a d)) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=\frac {b (b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac {2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac {b (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 91, normalized size = 0.82 \begin {gather*} \sqrt {c+d x^2} \left (-\frac {a^2}{3 x^3}-\frac {a (a d+6 b c)}{3 c x}+\frac {b^2 x}{2}\right )+\frac {b (4 a d+b c) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{2 \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 97, normalized size = 0.87 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-2 a^2 c-2 a^2 d x^2-12 a b c x^2+3 b^2 c x^4\right )}{6 c x^3}+\frac {\left (b^2 (-c)-4 a b d\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{2 \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.64, size = 210, normalized size = 1.89 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {d} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (3 \, b^{2} c d x^{4} - 2 \, a^{2} c d - 2 \, {\left (6 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, c d x^{3}}, -\frac {3 \, {\left (b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (3 \, b^{2} c d x^{4} - 2 \, a^{2} c d - 2 \, {\left (6 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, c d x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 188, normalized size = 1.69 \begin {gather*} \frac {1}{2} \, \sqrt {d x^{2} + c} b^{2} x - \frac {{\left (b^{2} c \sqrt {d} + 4 \, a b d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, d} + \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{2} \sqrt {d} + 6 \, a b c^{3} \sqrt {d} + a^{2} c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 122, normalized size = 1.10 \begin {gather*} 2 a b \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\frac {b^{2} c \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}+\frac {2 \sqrt {d \,x^{2}+c}\, a b d x}{c}+\frac {\sqrt {d \,x^{2}+c}\, b^{2} x}{2}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b}{c x}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2}}{3 c \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 86, normalized size = 0.77 \begin {gather*} \frac {1}{2} \, \sqrt {d x^{2} + c} b^{2} x + \frac {b^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {d}} + 2 \, a b \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {2 \, \sqrt {d x^{2} + c} a b}{x} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{3 \, c x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.46, size = 170, normalized size = 1.53 \begin {gather*} - \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3 c} - \frac {2 a b \sqrt {c}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + 2 a b \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {2 a b d x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {b^{2} \sqrt {c} x \sqrt {1 + \frac {d x^{2}}{c}}}{2} + \frac {b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{2 \sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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