Optimal. Leaf size=133 \[ -\frac {\left (b^2 c^2-8 a d (b c+a d)\right ) x \sqrt {c+d x^2}}{8 c d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d}-\frac {\left (b^2 c^2-8 a d (b c+a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 0.98, 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 = {473, 396, 201,
223, 212} \begin {gather*} -\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}-\frac {\left (b^2 c^2-8 a d (a d+b c)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2}}-\frac {1}{8} x \sqrt {c+d x^2} \left (\frac {b^2 c}{d}-\frac {8 a (a d+b c)}{c}\right )+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 396
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^2} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {\int \left (2 a (b c+a d)+b^2 c x^2\right ) \sqrt {c+d x^2} \, dx}{c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d}-\frac {1}{4} \left (\frac {b^2 c}{d}-\frac {8 a (b c+a d)}{c}\right ) \int \sqrt {c+d x^2} \, dx\\ &=-\frac {1}{8} \left (\frac {b^2 c}{d}-\frac {8 a (b c+a d)}{c}\right ) x \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d}-\frac {1}{8} \left (\frac {b^2 c^2}{d}-8 a (b c+a d)\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=-\frac {1}{8} \left (\frac {b^2 c}{d}-\frac {8 a (b c+a d)}{c}\right ) x \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d}-\frac {1}{8} \left (\frac {b^2 c^2}{d}-8 a (b c+a d)\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=-\frac {1}{8} \left (\frac {b^2 c}{d}-\frac {8 a (b c+a d)}{c}\right ) x \sqrt {c+d x^2}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{3/2}}{4 d}-\frac {\left (\frac {b^2 c^2}{d}-8 a (b c+a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 106, normalized size = 0.80 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-8 a^2 d+b^2 c x^2+8 a b d x^2+2 b^2 d x^4\right )}{8 d x}+\frac {\left (b^2 c^2-8 a b c d-8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{8 d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 165, normalized size = 1.24
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 b^{2} d \,x^{4}-8 a b d \,x^{2}-b^{2} c \,x^{2}+8 a^{2} d \right )}{8 d x}+\sqrt {d}\, \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b c}{\sqrt {d}}-\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2} c^{2}}{8 d^{\frac {3}{2}}}\) | \(125\) |
default | \(b^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4 d}-\frac {c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4 d}\right )+2 a b \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{c x}+\frac {2 d \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{c}\right )\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 120, normalized size = 0.90 \begin {gather*} \sqrt {d x^{2} + c} a b x + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x}{4 \, d} - \frac {\sqrt {d x^{2} + c} b^{2} c x}{8 \, d} - \frac {b^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {3}{2}}} + \frac {a b c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {d}} + a^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {\sqrt {d x^{2} + c} a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 215, normalized size = 1.62 \begin {gather*} \left [-\frac {{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{4} - 8 \, a^{2} d^{2} + {\left (b^{2} c d + 8 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{16 \, d^{2} x}, \frac {{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} d^{2} x^{4} - 8 \, a^{2} d^{2} + {\left (b^{2} c d + 8 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, d^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.17, size = 219, normalized size = 1.65 \begin {gather*} - \frac {a^{2} \sqrt {c}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + a^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {a^{2} d x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + a b \sqrt {c} x \sqrt {1 + \frac {d x^{2}}{c}} + \frac {a b c \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{\sqrt {d}} + \frac {b^{2} c^{\frac {3}{2}} x}{8 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 b^{2} \sqrt {c} x^{3}}{8 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {3}{2}}} + \frac {b^{2} d x^{5}}{4 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 126, normalized size = 0.95 \begin {gather*} \frac {2 \, a^{2} c \sqrt {d}}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} + \frac {1}{8} \, {\left (2 \, b^{2} x^{2} + \frac {b^{2} c d + 8 \, a b d^{2}}{d^{2}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (b^{2} c^{2} \sqrt {d} - 8 \, a b c d^{\frac {3}{2}} - 8 \, a^{2} d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{16 \, d^{2}} \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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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