Optimal. Leaf size=222 \[ -\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {\left (c+d x^2\right )^{5/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c^2 x^2}+\frac {5 d \left (c+d x^2\right )^{3/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{48 c^2}+\frac {5 d \sqrt {c+d x^2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c}-\frac {5 d \left (a d (a d+12 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}}-\frac {a \left (c+d x^2\right )^{7/2} (a d+12 b c)}{24 c^2 x^4} \]
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Rubi [A] time = 0.25, antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {446, 89, 78, 47, 50, 63, 208} \[ -\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {\left (c+d x^2\right )^{5/2} \left (\frac {a d (a d+12 b c)}{c^2}+8 b^2\right )}{16 x^2}+\frac {5 d \left (c+d x^2\right )^{3/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{48 c^2}+\frac {5 d \sqrt {c+d x^2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c}-\frac {5 d \left (a d (a d+12 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}}-\frac {a \left (c+d x^2\right )^{7/2} (a d+12 b c)}{24 c^2 x^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}+\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {1}{2} a (12 b c+a d)+3 b^2 c x\right ) (c+d x)^{5/2}}{x^3} \, dx,x,x^2\right )}{6 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{16} \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{5/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 d \left (8 b^2 c^2+12 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{16} \left (5 \left (8 b^2 c^2+12 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )\\ &=\frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}-\frac {5 d \left (8 b^2 c^2+12 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 92, normalized size = 0.41 \[ \frac {\left (c+d x^2\right )^{7/2} \left (3 d x^6 \left (a^2 d^2+12 a b c d+8 b^2 c^2\right ) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {d x^2}{c}+1\right )-7 a c^2 \left (4 a c+a d x^2+12 b c x^2\right )\right )}{168 c^4 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 347, normalized size = 1.56 \[ \left [\frac {15 \, {\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {c} x^{6} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (16 \, b^{2} c d^{2} x^{8} + 16 \, {\left (7 \, b^{2} c^{2} d + 6 \, a b c d^{2}\right )} x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 36 \, a b c^{2} d + 11 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 13 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, c x^{6}}, \frac {15 \, {\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (16 \, b^{2} c d^{2} x^{8} + 16 \, {\left (7 \, b^{2} c^{2} d + 6 \, a b c d^{2}\right )} x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 36 \, a b c^{2} d + 11 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 13 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, c x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 286, normalized size = 1.29 \[ \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2} + 96 \, \sqrt {d x^{2} + c} b^{2} c d^{2} + 96 \, \sqrt {d x^{2} + c} a b d^{3} + \frac {15 \, {\left (8 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} + a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d^{2} + 108 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{3} - 192 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} + 84 \, \sqrt {d x^{2} + c} a b c^{3} d^{3} + 33 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{4} - 40 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{4} + 15 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{4}}{d^{3} x^{6}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 387, normalized size = 1.74 \[ -\frac {5 a^{2} d^{3} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{16 \sqrt {c}}-\frac {15 a b \sqrt {c}\, d^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{4}-\frac {5 b^{2} c^{\frac {3}{2}} d \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{2}+\frac {5 \sqrt {d \,x^{2}+c}\, a^{2} d^{3}}{16 c}+\frac {15 \sqrt {d \,x^{2}+c}\, a b \,d^{2}}{4}+\frac {5 \sqrt {d \,x^{2}+c}\, b^{2} c d}{2}+\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} d^{3}}{48 c^{2}}+\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,d^{2}}{4 c}+\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} d}{6}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2} d^{3}}{16 c^{3}}+\frac {3 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a b \,d^{2}}{4 c^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} d}{2 c}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2} d^{2}}{16 c^{3} x^{2}}-\frac {3 \left (d \,x^{2}+c \right )^{\frac {7}{2}} a b d}{4 c^{2} x^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} b^{2}}{2 c \,x^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2} d}{24 c^{2} x^{4}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a b}{2 c \,x^{4}}-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} a^{2}}{6 c \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 353, normalized size = 1.59 \[ -\frac {5}{2} \, b^{2} c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {15}{4} \, a b \sqrt {c} d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {5 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, \sqrt {c}} + \frac {5}{6} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d}{2 \, c} + \frac {5}{2} \, \sqrt {d x^{2} + c} b^{2} c d + \frac {15}{4} \, \sqrt {d x^{2} + c} a b d^{2} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{2}}{4 \, c^{2}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{4 \, c} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{3}}{16 \, c^{3}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{48 \, c^{2}} + \frac {5 \, \sqrt {d x^{2} + c} a^{2} d^{3}}{16 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2}}{2 \, c x^{2}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d}{4 \, c^{2} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{16 \, c^{3} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{2 \, c x^{4}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{24 \, c^{2} x^{4}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{6 \, c x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.72, size = 301, normalized size = 1.36 \[ \frac {\sqrt {d\,x^2+c}\,\left (\frac {5\,a^2\,c^2\,d^3}{16}+\frac {7\,a\,b\,c^3\,d^2}{4}+\frac {b^2\,c^4\,d}{2}\right )-{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {5\,a^2\,c\,d^3}{6}+4\,a\,b\,c^2\,d^2+b^2\,c^3\,d\right )+{\left (d\,x^2+c\right )}^{5/2}\,\left (\frac {11\,a^2\,d^3}{16}+\frac {9\,a\,b\,c\,d^2}{4}+\frac {b^2\,c^2\,d}{2}\right )}{3\,c\,{\left (d\,x^2+c\right )}^2-3\,c^2\,\left (d\,x^2+c\right )-{\left (d\,x^2+c\right )}^3+c^3}+\left (2\,b\,d\,\left (a\,d-b\,c\right )+4\,b^2\,c\,d\right )\,\sqrt {d\,x^2+c}+\frac {b^2\,d\,{\left (d\,x^2+c\right )}^{3/2}}{3}+\frac {d\,\mathrm {atan}\left (\frac {d\,\sqrt {d\,x^2+c}\,\left (a^2\,d^2+12\,a\,b\,c\,d+8\,b^2\,c^2\right )\,5{}\mathrm {i}}{8\,\sqrt {c}\,\left (\frac {5\,a^2\,d^3}{8}+\frac {15\,a\,b\,c\,d^2}{2}+5\,b^2\,c^2\,d\right )}\right )\,\left (a^2\,d^2+12\,a\,b\,c\,d+8\,b^2\,c^2\right )\,5{}\mathrm {i}}{16\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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