Optimal. Leaf size=228 \[ -\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {\left (c+d x^2\right )^{5/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{15 c^2 x}+\frac {d x \left (c+d x^2\right )^{3/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{12 c^2}+\frac {d x \sqrt {c+d x^2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{8 c}+\frac {1}{8} \sqrt {d} \left (8 a d (a d+5 b c)+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )-\frac {2 a \left (c+d x^2\right )^{7/2} (a d+5 b c)}{15 c^2 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 225, normalized size of antiderivative = 0.99, number of steps used = 7, 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 )^{7/2}}{5 c x^5}-\frac {\left (c+d x^2\right )^{5/2} \left (\frac {8 a d (a d+5 b c)}{c^2}+15 b^2\right )}{15 x}+\frac {d x \left (c+d x^2\right )^{3/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{12 c^2}+\frac {d x \sqrt {c+d x^2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{8 c}+\frac {1}{8} \sqrt {d} \left (8 a d (a d+5 b c)+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )-\frac {2 a \left (c+d x^2\right )^{7/2} (a d+5 b c)}{15 c^2 x^3} \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 )^{5/2}}{x^6} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}+\frac {\int \frac {\left (2 a (5 b c+a d)+5 b^2 c x^2\right ) \left (c+d x^2\right )^{5/2}}{x^4} \, dx}{5 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}-\frac {1}{15} \left (-15 b^2-\frac {8 a d (5 b c+a d)}{c^2}\right ) \int \frac {\left (c+d x^2\right )^{5/2}}{x^2} \, dx\\ &=-\frac {\left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{15 x}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}+\frac {1}{3} \left (d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {1}{12} d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}-\frac {\left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{15 x}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}+\frac {1}{4} \left (c d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right )\right ) \int \sqrt {c+d x^2} \, dx\\ &=\frac {1}{8} c d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \sqrt {c+d x^2}+\frac {1}{12} d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}-\frac {\left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{15 x}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}+\frac {1}{8} \left (d \left (15 b^2 c^2+40 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=\frac {1}{8} c d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \sqrt {c+d x^2}+\frac {1}{12} d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}-\frac {\left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{15 x}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}+\frac {1}{8} \left (d \left (15 b^2 c^2+40 a b c d+8 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=\frac {1}{8} c d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \sqrt {c+d x^2}+\frac {1}{12} d \left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}-\frac {\left (15 b^2+\frac {8 a d (5 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{15 x}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac {2 a (5 b c+a d) \left (c+d x^2\right )^{7/2}}{15 c^2 x^3}+\frac {1}{8} \sqrt {d} \left (15 b^2 c^2+40 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 158, normalized size = 0.69 \begin {gather*} \frac {1}{8} \sqrt {d} \left (8 a^2 d^2+40 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+\sqrt {c+d x^2} \left (\frac {-23 a^2 d^2-70 a b c d-15 b^2 c^2}{15 x}-\frac {a^2 c^2}{5 x^5}-\frac {a c (11 a d+10 b c)}{15 x^3}+\frac {1}{8} b d x (8 a d+9 b c)+\frac {1}{4} b^2 d^2 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 169, normalized size = 0.74 \begin {gather*} \frac {1}{8} \left (-8 a^2 d^{5/2}-40 a b c d^{3/2}-15 b^2 c^2 \sqrt {d}\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )+\frac {\sqrt {c+d x^2} \left (-24 a^2 c^2-88 a^2 c d x^2-184 a^2 d^2 x^4-80 a b c^2 x^2-560 a b c d x^4+120 a b d^2 x^6-120 b^2 c^2 x^4+135 b^2 c d x^6+30 b^2 d^2 x^8\right )}{120 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 318, normalized size = 1.39 \begin {gather*} \left [\frac {15 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (30 \, b^{2} d^{2} x^{8} + 15 \, {\left (9 \, b^{2} c d + 8 \, a b d^{2}\right )} x^{6} - 8 \, {\left (15 \, b^{2} c^{2} + 70 \, a b c d + 23 \, a^{2} d^{2}\right )} x^{4} - 24 \, a^{2} c^{2} - 8 \, {\left (10 \, a b c^{2} + 11 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{240 \, x^{5}}, -\frac {15 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (30 \, b^{2} d^{2} x^{8} + 15 \, {\left (9 \, b^{2} c d + 8 \, a b d^{2}\right )} x^{6} - 8 \, {\left (15 \, b^{2} c^{2} + 70 \, a b c d + 23 \, a^{2} d^{2}\right )} x^{4} - 24 \, a^{2} c^{2} - 8 \, {\left (10 \, a b c^{2} + 11 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{120 \, x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 510, normalized size = 2.24 \begin {gather*} \frac {1}{8} \, {\left (2 \, b^{2} d^{2} x^{2} + \frac {9 \, b^{2} c d^{3} + 8 \, a b d^{4}}{d^{2}}\right )} \sqrt {d x^{2} + c} x - \frac {1}{16} \, {\left (15 \, b^{2} c^{2} \sqrt {d} + 40 \, 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 ) + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{3} \sqrt {d} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac {3}{2}} + 45 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{4} \sqrt {d} - 300 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac {3}{2}} - 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac {5}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{5} \sqrt {d} + 400 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac {3}{2}} + 140 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{6} \sqrt {d} - 260 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac {3}{2}} - 70 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac {5}{2}} + 15 \, b^{2} c^{7} \sqrt {d} + 70 \, a b c^{6} d^{\frac {3}{2}} + 23 \, a^{2} c^{5} 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 = 369, normalized size = 1.62 \begin {gather*} a^{2} d^{\frac {5}{2}} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+5 a b c \,d^{\frac {3}{2}} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\frac {15 b^{2} c^{2} \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8}+\frac {\sqrt {d \,x^{2}+c}\, a^{2} d^{3} x}{c}+5 \sqrt {d \,x^{2}+c}\, a b \,d^{2} x +\frac {15 \sqrt {d \,x^{2}+c}\, b^{2} c d x}{8}+\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} d^{3} x}{3 c^{2}}+\frac {10 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,d^{2} x}{3 c}+\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} d x}{4}+\frac {8 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2} d^{3} x}{15 c^{3}}+\frac {8 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a b \,d^{2} x}{3 c^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} d x}{c}-\frac {8 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2} d^{2}}{15 c^{3} x}-\frac {8 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a b d}{3 c^{2} x}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} b^{2}}{c x}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2} d}{15 c^{2} x^{3}}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a b}{3 c \,x^{3}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{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 = 0.96, size = 285, normalized size = 1.25 \begin {gather*} \frac {5}{4} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d x + \frac {15}{8} \, \sqrt {d x^{2} + c} b^{2} c d x + 5 \, \sqrt {d x^{2} + c} a b d^{2} x + \frac {10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2} x}{3 \, c} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3} x}{3 \, c^{2}} + \frac {\sqrt {d x^{2} + c} a^{2} d^{3} x}{c} + \frac {15}{8} \, b^{2} c^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + 5 \, a b c d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + a^{2} d^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2}}{x} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d}{3 \, c x} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{15 \, c^{2} x} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{3 \, c x^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{15 \, c^{2} x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{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 = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.41, size = 474, normalized size = 2.08 \begin {gather*} - \frac {a^{2} \sqrt {c} d^{2}}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {11 a^{2} c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 x^{2}} - \frac {8 a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15} + a^{2} d^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {a^{2} d^{3} x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {4 a b c^{\frac {3}{2}} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} + a b \sqrt {c} d^{2} x \sqrt {1 + \frac {d x^{2}}{c}} - \frac {4 a b \sqrt {c} d^{2} x}{\sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a b c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {2 a b c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + 5 a b c d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {b^{2} c^{\frac {5}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + b^{2} c^{\frac {3}{2}} d x \sqrt {1 + \frac {d x^{2}}{c}} - \frac {7 b^{2} c^{\frac {3}{2}} d x}{8 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 b^{2} \sqrt {c} d^{2} x^{3}}{8 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {15 b^{2} c^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8} + \frac {b^{2} d^{3} 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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