Optimal. Leaf size=175 \[ -\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) x \sqrt {c+d x^2}}{16 d}-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) x \left (c+d x^2\right )^{3/2}}{24 c d}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {c \left (b^2 c^2-12 a d (b c+2 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 172, normalized size of antiderivative = 0.98, number of steps
used = 6, 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 )^{5/2}}{c x}-\frac {c \left (b^2 c^2-12 a d (2 a d+b c)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{3/2}}-\frac {x \sqrt {c+d x^2} \left (b^2 c^2-12 a d (2 a d+b c)\right )}{16 d}-\frac {1}{24} x \left (c+d x^2\right )^{3/2} \left (\frac {b^2 c}{d}-\frac {12 a (2 a d+b c)}{c}\right )+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 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 \left (c+d x^2\right )^{3/2}}{x^2} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {\int \left (2 a (b c+2 a d)+b^2 c x^2\right ) \left (c+d x^2\right )^{3/2} \, dx}{c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) \int \left (c+d x^2\right )^{3/2} \, dx}{6 c d}\\ &=-\frac {1}{24} \left (\frac {b^2 c}{d}-\frac {12 a (b c+2 a d)}{c}\right ) x \left (c+d x^2\right )^{3/2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) \int \sqrt {c+d x^2} \, dx}{8 d}\\ &=-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) x \sqrt {c+d x^2}}{16 d}-\frac {1}{24} \left (\frac {b^2 c}{d}-\frac {12 a (b c+2 a d)}{c}\right ) x \left (c+d x^2\right )^{3/2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {\left (c \left (b^2 c^2-12 a d (b c+2 a d)\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 d}\\ &=-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) x \sqrt {c+d x^2}}{16 d}-\frac {1}{24} \left (\frac {b^2 c}{d}-\frac {12 a (b c+2 a d)}{c}\right ) x \left (c+d x^2\right )^{3/2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {\left (c \left (b^2 c^2-12 a d (b c+2 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 d}\\ &=-\frac {\left (b^2 c^2-12 a d (b c+2 a d)\right ) x \sqrt {c+d x^2}}{16 d}-\frac {1}{24} \left (\frac {b^2 c}{d}-\frac {12 a (b c+2 a d)}{c}\right ) x \left (c+d x^2\right )^{3/2}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{5/2}}{6 d}-\frac {c \left (b^2 c^2-12 a d (b c+2 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 139, normalized size = 0.79 \begin {gather*} \frac {\sqrt {d} \sqrt {c+d x^2} \left (24 a^2 d \left (-2 c+d x^2\right )+12 a b d x^2 \left (5 c+2 d x^2\right )+b^2 x^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )+3 c \left (b^2 c^2-12 a b c d-24 a^2 d^2\right ) x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{48 d^{3/2} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 213, normalized size = 1.22
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-8 b^{2} d^{2} x^{6}-24 a b \,d^{2} x^{4}-14 b^{2} c d \,x^{4}-24 a^{2} d^{2} x^{2}-60 a b c d \,x^{2}-3 b^{2} c^{2} x^{2}+48 a^{2} c d \right )}{48 d x}+\frac {3 c \sqrt {d}\, \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{2}+\frac {3 c^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b}{4 \sqrt {d}}-\frac {c^{3} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{16 d^{\frac {3}{2}}}\) | \(167\) |
default | \(b^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )+2 a b \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 178, normalized size = 1.02 \begin {gather*} \frac {1}{2} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b x + \frac {3}{4} \, \sqrt {d x^{2} + c} a b c x + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x}{6 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x}{24 \, d} - \frac {\sqrt {d x^{2} + c} b^{2} c^{2} x}{16 \, d} + \frac {3}{2} \, \sqrt {d x^{2} + c} a^{2} d x - \frac {b^{2} c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, d^{\frac {3}{2}}} + \frac {3 \, a b c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{4 \, \sqrt {d}} + \frac {3}{2} \, a^{2} c \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 293, normalized size = 1.67 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} c^{3} - 12 \, a b c^{2} d - 24 \, a^{2} c 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 (8 \, b^{2} d^{3} x^{6} - 48 \, a^{2} c d^{2} + 2 \, {\left (7 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{4} + 3 \, {\left (b^{2} c^{2} d + 20 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, d^{2} x}, \frac {3 \, {\left (b^{2} c^{3} - 12 \, a b c^{2} d - 24 \, a^{2} c d^{2}\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (8 \, b^{2} d^{3} x^{6} - 48 \, a^{2} c d^{2} + 2 \, {\left (7 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{4} + 3 \, {\left (b^{2} c^{2} d + 20 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, d^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs.
\(2 (156) = 312\).
time = 12.77, size = 367, normalized size = 2.10 \begin {gather*} - \frac {a^{2} c^{\frac {3}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a^{2} \sqrt {c} d x \sqrt {1 + \frac {d x^{2}}{c}}}{2} - \frac {a^{2} \sqrt {c} d x}{\sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{2} + a b c^{\frac {3}{2}} x \sqrt {1 + \frac {d x^{2}}{c}} + \frac {a b c^{\frac {3}{2}} x}{4 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b \sqrt {c} d x^{3}}{4 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{4 \sqrt {d}} + \frac {a b d^{2} x^{5}}{2 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {b^{2} c^{\frac {5}{2}} x}{16 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {17 b^{2} c^{\frac {3}{2}} x^{3}}{48 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {11 b^{2} \sqrt {c} d x^{5}}{24 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{16 d^{\frac {3}{2}}} + \frac {b^{2} d^{2} x^{7}}{6 \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 = 1.63, size = 173, normalized size = 0.99 \begin {gather*} \frac {2 \, a^{2} c^{2} \sqrt {d}}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} + \frac {1}{48} \, {\left (2 \, {\left (4 \, b^{2} d x^{2} + \frac {7 \, b^{2} c d^{4} + 12 \, a b d^{5}}{d^{4}}\right )} x^{2} + \frac {3 \, {\left (b^{2} c^{2} d^{3} + 20 \, a b c d^{4} + 8 \, a^{2} d^{5}\right )}}{d^{4}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (b^{2} c^{3} \sqrt {d} - 12 \, a b c^{2} d^{\frac {3}{2}} - 24 \, a^{2} c d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{32 \, 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\,{\left (d\,x^2+c\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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