Optimal. Leaf size=223 \[ \frac {5}{16} \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \sqrt {c+d x^2}+\frac {5 \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \left (c+d x^2\right )^{3/2}}{24 c}+\frac {\left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \left (c+d x^2\right )^{5/2}}{6 c^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {5 c \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}} \]
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Rubi [A]
time = 0.12, antiderivative size = 219, normalized size of antiderivative = 0.98, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {473, 464, 201,
223, 212} \begin {gather*} -\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac {1}{6} x \left (c+d x^2\right )^{5/2} \left (\frac {4 a d (2 a d+3 b c)}{c^2}+b^2\right )+\frac {5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac {5}{16} x \sqrt {c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )+\frac {5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}}-\frac {2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 464
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac {\int \frac {\left (2 a (3 b c+2 a d)+3 b^2 c x^2\right ) \left (c+d x^2\right )^{5/2}}{x^2} \, dx}{3 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) \int \left (c+d x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{6} \left (5 c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{8} \left (5 \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \sqrt {c+d x^2} \, dx\\ &=\frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=\frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=\frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 154, normalized size = 0.69 \begin {gather*} \frac {1}{48} \left (\frac {\sqrt {c+d x^2} \left (-8 a^2 \left (2 c^2+14 c d x^2-3 d^2 x^4\right )+12 a b x^2 \left (-8 c^2+9 c d x^2+2 d^2 x^4\right )+b^2 x^4 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )}{x^3}-\frac {15 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{\sqrt {d}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 287, normalized size = 1.29
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-8 b^{2} d^{2} x^{8}-24 a b \,d^{2} x^{6}-26 b^{2} c d \,x^{6}-24 a^{2} d^{2} x^{4}-108 a b c d \,x^{4}-33 b^{2} c^{2} x^{4}+112 a^{2} c d \,x^{2}+96 a b \,c^{2} x^{2}+16 a^{2} c^{2}\right )}{48 x^{3}}+\frac {5 c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) d^{\frac {3}{2}} a^{2}}{2}+\frac {15 c^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) \sqrt {d}\, a b}{4}+\frac {5 c^{3} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{16 \sqrt {d}}\) | \(185\) |
default | \(b^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{3 c \,x^{3}}+\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{c x}+\frac {6 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{c}\right )}{3 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{c x}+\frac {6 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{c}\right )\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 234, normalized size = 1.05 \begin {gather*} \frac {1}{6} \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x + \frac {5}{24} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x + \frac {5}{16} \, \sqrt {d x^{2} + c} b^{2} c^{2} x + \frac {5}{2} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d x + \frac {15}{4} \, \sqrt {d x^{2} + c} a b c d x + \frac {5}{2} \, \sqrt {d x^{2} + c} a^{2} d^{2} x + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2} x}{3 \, c} + \frac {5 \, b^{2} c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, \sqrt {d}} + \frac {15}{4} \, a b c^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + \frac {5}{2} \, a^{2} c d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{x} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{3 \, c x} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{3 \, c x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.21, size = 346, normalized size = 1.55 \begin {gather*} \left [\frac {15 \, {\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {d} x^{3} \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^{8} + 2 \, {\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \, {\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \, {\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, d x^{3}}, -\frac {15 \, {\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (8 \, b^{2} d^{3} x^{8} + 2 \, {\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \, {\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \, {\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, d x^{3}}\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. 490 vs.
\(2 (221) = 442\).
time = 15.15, size = 490, normalized size = 2.20 \begin {gather*} - \frac {2 a^{2} c^{\frac {3}{2}} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a^{2} \sqrt {c} d^{2} x \sqrt {1 + \frac {d x^{2}}{c}}}{2} - \frac {2 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}}{3 x^{2}} - \frac {a^{2} c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + \frac {5 a^{2} c d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{2} - \frac {2 a b c^{\frac {5}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + 2 a b c^{\frac {3}{2}} d x \sqrt {1 + \frac {d x^{2}}{c}} - \frac {7 a b c^{\frac {3}{2}} d x}{4 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b \sqrt {c} d^{2} x^{3}}{4 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {15 a b c^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{4} + \frac {a b d^{3} x^{5}}{2 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {b^{2} c^{\frac {5}{2}} x \sqrt {1 + \frac {d x^{2}}{c}}}{2} + \frac {3 b^{2} c^{\frac {5}{2}} x}{16 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {35 b^{2} c^{\frac {3}{2}} d x^{3}}{48 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {17 b^{2} \sqrt {c} d^{2} x^{5}}{24 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{16 \sqrt {d}} + \frac {b^{2} d^{3} 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 = 0.69, size = 307, normalized size = 1.38 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (4 \, b^{2} d^{2} x^{2} + \frac {13 \, b^{2} c d^{5} + 12 \, a b d^{6}}{d^{4}}\right )} x^{2} + \frac {3 \, {\left (11 \, b^{2} c^{2} d^{4} + 36 \, a b c d^{5} + 8 \, a^{2} d^{6}\right )}}{d^{4}}\right )} \sqrt {d x^{2} + c} x - \frac {5 \, {\left (b^{2} c^{3} \sqrt {d} + 12 \, a b c^{2} d^{\frac {3}{2}} + 8 \, a^{2} c d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{32 \, d} + \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} \sqrt {d} + 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac {3}{2}} + 6 \, a b c^{5} \sqrt {d} + 7 \, a^{2} c^{4} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3}} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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