Optimal. Leaf size=147 \[ -\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {b \left (c+d x^2\right )^{3/2} (4 a d+3 b c)}{3 c x}+\frac {b d x \sqrt {c+d x^2} (4 a d+3 b c)}{2 c}+\frac {1}{2} b \sqrt {d} (4 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, 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 = {462, 453, 277, 195, 217, 206} \begin {gather*} -\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}-\frac {b \left (c+d x^2\right )^{3/2} (4 a d+3 b c)}{3 c x}+\frac {b d x \sqrt {c+d x^2} (4 a d+3 b c)}{2 c}+\frac {1}{2} b \sqrt {d} (4 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 277
Rule 453
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}+\frac {\int \frac {\left (10 a b c+5 b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{x^4} \, dx}{5 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {(b (3 b c+4 a d)) \int \frac {\left (c+d x^2\right )^{3/2}}{x^2} \, dx}{3 c}\\ &=-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {(b d (3 b c+4 a d)) \int \sqrt {c+d x^2} \, dx}{c}\\ &=\frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} (b d (3 b c+4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=\frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} (b d (3 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 d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 113, normalized size = 0.77 \begin {gather*} \frac {1}{2} b \sqrt {d} (4 a d+3 b c) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )-\frac {\sqrt {c+d x^2} \left (6 a^2 \left (c+d x^2\right )^2+20 a b c x^2 \left (c+4 d x^2\right )+15 b^2 c x^4 \left (2 c-d x^2\right )\right )}{30 c x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 138, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-6 a^2 c^2-12 a^2 c d x^2-6 a^2 d^2 x^4-20 a b c^2 x^2-80 a b c d x^4-30 b^2 c^2 x^4+15 b^2 c d x^6\right )}{30 c x^5}+\frac {1}{2} \left (-4 a b d^{3/2}-3 b^2 c \sqrt {d}\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 266, normalized size = 1.81 \begin {gather*} \left [\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, c x^{5}}, -\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, c x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 407, normalized size = 2.77 \begin {gather*} \frac {1}{2} \, \sqrt {d x^{2} + c} b^{2} d x - \frac {1}{4} \, {\left (3 \, 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 ) + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt {d} + 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c d^{\frac {3}{2}} + 15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt {d} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac {3}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt {d} + 220 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt {d} - 140 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac {3}{2}} + 15 \, b^{2} c^{6} \sqrt {d} + 40 \, a b c^{5} d^{\frac {3}{2}} + 3 \, a^{2} c^{4} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 203, normalized size = 1.38 \begin {gather*} 2 a b \,d^{\frac {3}{2}} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\frac {3 b^{2} c \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{2}+\frac {2 \sqrt {d \,x^{2}+c}\, a b \,d^{2} x}{c}+\frac {3 \sqrt {d \,x^{2}+c}\, b^{2} d x}{2}+\frac {4 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,d^{2} x}{3 c^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} d x}{c}-\frac {4 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a b d}{3 c^{2} x}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2}}{c x}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a b}{3 c \,x^{3}}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2}}{5 c \,x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 147, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, \sqrt {d x^{2} + c} b^{2} d x + \frac {2 \, \sqrt {d x^{2} + c} a b d^{2} x}{c} + \frac {3}{2} \, b^{2} c \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + 2 \, a b d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{x} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{3 \, c x} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{3 \, c x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{5 \, c x^{5}} \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^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.79, size = 304, normalized size = 2.07 \begin {gather*} - \frac {a^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {2 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{2}} - \frac {a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 c} - \frac {2 a b \sqrt {c} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a b c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {2 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + 2 a b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {2 a b d^{2} x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {3}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {b^{2} \sqrt {c} d x \sqrt {1 + \frac {d x^{2}}{c}}}{2} - \frac {b^{2} \sqrt {c} d x}{\sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 b^{2} c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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