Optimal. Leaf size=222 \[ \frac {1}{8} \left (8 b^2 c^2+5 a d (8 b c+3 a d)\right ) \sqrt {c+d x^2}+\frac {\left (8 b^2 c^2+5 a d (8 b c+3 a d)\right ) \left (c+d x^2\right )^{3/2}}{24 c}+\frac {\left (8 b^2 c^2+5 a d (8 b c+3 a d)\right ) \left (c+d x^2\right )^{5/2}}{40 c^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}-\frac {1}{8} \sqrt {c} \left (8 b^2 c^2+5 a d (8 b c+3 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 91, 79, 52,
65, 214} \begin {gather*} -\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}+\frac {1}{40} \left (c+d x^2\right )^{5/2} \left (\frac {5 a d (3 a d+8 b c)}{c^2}+8 b^2\right )+\frac {\left (c+d x^2\right )^{3/2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )}{24 c}+\frac {1}{8} \sqrt {c+d x^2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )-\frac {1}{8} \sqrt {c} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )-\frac {a \left (c+d x^2\right )^{7/2} (3 a d+8 b c)}{8 c^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 91
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^{5/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} a (8 b c+3 a d)+2 b^2 c x\right ) (c+d x)^{5/2}}{x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}+\frac {1}{16} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{40} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}+\frac {1}{16} \left (c \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{24} c \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}+\frac {1}{40} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}+\frac {1}{16} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{8} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \sqrt {c+d x^2}+\frac {1}{24} c \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}+\frac {1}{40} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}+\frac {1}{16} \left (c \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {1}{8} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \sqrt {c+d x^2}+\frac {1}{24} c \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}+\frac {1}{40} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}+\frac {\left (c \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{8 d}\\ &=\frac {1}{8} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \sqrt {c+d x^2}+\frac {1}{24} c \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}+\frac {1}{40} \left (8 b^2+\frac {5 a d (8 b c+3 a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}-\frac {a (8 b c+3 a d) \left (c+d x^2\right )^{7/2}}{8 c^2 x^2}-\frac {1}{8} \sqrt {c} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 153, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-15 a^2 \left (2 c^2+9 c d x^2-8 d^2 x^4\right )+40 a b x^2 \left (-3 c^2+14 c d x^2+2 d^2 x^4\right )+8 b^2 x^4 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )}{120 x^4}-\frac {1}{8} \sqrt {c} \left (8 b^2 c^2+40 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 284, normalized size = 1.28
method | result | size |
risch | \(-\frac {c a \sqrt {d \,x^{2}+c}\, \left (9 a d \,x^{2}+8 c \,x^{2} b +2 a c \right )}{8 x^{4}}+\frac {b^{2} d^{2} x^{4} \sqrt {d \,x^{2}+c}}{5}+\frac {11 b^{2} d c \,x^{2} \sqrt {d \,x^{2}+c}}{15}+\frac {23 b^{2} c^{2} \sqrt {d \,x^{2}+c}}{15}+\frac {2 a b \,d^{2} x^{2} \sqrt {d \,x^{2}+c}}{3}+\frac {14 a b c d \sqrt {d \,x^{2}+c}}{3}+a^{2} d^{2} \sqrt {d \,x^{2}+c}-\frac {15 \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a^{2} d^{2}}{8}-5 c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a b d -c^{\frac {5}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b^{2}\) | \(239\) |
default | \(a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{4 c \,x^{4}}+\frac {3 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )}{4 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )+b^{2} \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 271, normalized size = 1.22 \begin {gather*} -b^{2} c^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - 5 \, a b c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {15}{8} \, a^{2} \sqrt {c} d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) + \frac {1}{5} \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} + \frac {1}{3} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c + \sqrt {d x^{2} + c} b^{2} c^{2} + \frac {5}{3} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d}{c} + 5 \, \sqrt {d x^{2} + c} a b c d + \frac {15}{8} \, \sqrt {d x^{2} + c} a^{2} d^{2} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{8 \, c^{2}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{8 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{c x^{2}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{8 \, c^{2} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{4 \, c x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 319, normalized size = 1.44 \begin {gather*} \left [\frac {15 \, {\left (8 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {c} x^{4} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, b^{2} d^{2} x^{8} + 8 \, {\left (11 \, b^{2} c d + 10 \, a b d^{2}\right )} x^{6} + 8 \, {\left (23 \, b^{2} c^{2} + 70 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 30 \, a^{2} c^{2} - 15 \, {\left (8 \, a b c^{2} + 9 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{240 \, x^{4}}, \frac {15 \, {\left (8 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (24 \, b^{2} d^{2} x^{8} + 8 \, {\left (11 \, b^{2} c d + 10 \, a b d^{2}\right )} x^{6} + 8 \, {\left (23 \, b^{2} c^{2} + 70 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 30 \, a^{2} c^{2} - 15 \, {\left (8 \, a b c^{2} + 9 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{120 \, x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 84.67, size = 473, normalized size = 2.13 \begin {gather*} - \frac {15 a^{2} \sqrt {c} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{8} - \frac {a^{2} c^{3}}{4 \sqrt {d} x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a^{2} c^{2} \sqrt {d}}{8 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a^{2} c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{x} + \frac {7 a^{2} c d^{\frac {3}{2}}}{8 x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {a^{2} d^{\frac {5}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} - 5 a b c^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )} - \frac {a b c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{x} + \frac {4 a b c^{2} \sqrt {d}}{x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {4 a b c d^{\frac {3}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + 2 a b d^{2} \left (\begin {cases} \frac {\sqrt {c} x^{2}}{2} & \text {for}\: d = 0 \\\frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) - b^{2} c^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )} + \frac {b^{2} c^{3}}{\sqrt {d} x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {b^{2} c^{2} \sqrt {d} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + 2 b^{2} c d \left (\begin {cases} \frac {\sqrt {c} x^{2}}{2} & \text {for}\: d = 0 \\\frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + b^{2} d^{2} \left (\begin {cases} - \frac {2 c^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {c x^{2} \sqrt {c + d x^{2}}}{15 d} + \frac {x^{4} \sqrt {c + d x^{2}}}{5} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 242, normalized size = 1.09 \begin {gather*} \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d + 40 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c d + 120 \, \sqrt {d x^{2} + c} b^{2} c^{2} d + 80 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2} + 480 \, \sqrt {d x^{2} + c} a b c d^{2} + 120 \, \sqrt {d x^{2} + c} a^{2} d^{3} + \frac {15 \, {\left (8 \, b^{2} c^{3} d + 40 \, a b c^{2} d^{2} + 15 \, a^{2} c d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {15 \, {\left (8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 8 \, \sqrt {d x^{2} + c} a b c^{3} d^{2} + 9 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{3} - 7 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{3}\right )}}{d^{2} x^{4}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.88, size = 262, normalized size = 1.18 \begin {gather*} {\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {c\,b^2}{3}+\frac {2\,a\,d\,b}{3}\right )-\frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {9\,a^2\,c\,d^2}{8}+b\,a\,c^2\,d\right )-\left (\frac {7\,a^2\,c^2\,d^2}{8}+b\,a\,c^3\,d\right )\,\sqrt {d\,x^2+c}}{{\left (d\,x^2+c\right )}^2-2\,c\,\left (d\,x^2+c\right )+c^2}+\sqrt {d\,x^2+c}\,\left ({\left (a\,d-b\,c\right )}^2+3\,c\,\left (c\,b^2+2\,a\,d\,b\right )-3\,b^2\,c^2\right )+\frac {b^2\,{\left (d\,x^2+c\right )}^{5/2}}{5}+2\,\mathrm {atan}\left (\frac {2\,\sqrt {d\,x^2+c}\,\sqrt {-\frac {c}{256}}\,\left (15\,a^2\,d^2+40\,a\,b\,c\,d+8\,b^2\,c^2\right )}{\frac {15\,a^2\,c\,d^2}{8}+5\,a\,b\,c^2\,d+b^2\,c^3}\right )\,\sqrt {-\frac {c}{256}}\,\left (15\,a^2\,d^2+40\,a\,b\,c\,d+8\,b^2\,c^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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