Optimal. Leaf size=149 \[ -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a^2 c^2 x^2}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{5/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 105, 156,
12, 95, 214} \begin {gather*} \frac {3 \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{8 a^2 c^2 x^2}-\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{5/2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 156
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}-\frac {\text {Subst}\left (\int \frac {\frac {3}{2} (b c+a d)+b d x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a c}\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a^2 c^2 x^2}+\frac {\text {Subst}\left (\int \frac {3 b^2 c^2+2 a b c d+3 a^2 d^2}{4 x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 c^2}\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a^2 c^2 x^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 a^2 c^2}\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a^2 c^2 x^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{8 a^2 c^2}\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a^2 c^2 x^2}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.39, size = 126, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a c+3 b c x^2+3 a d x^2\right )}{8 a^2 c^2 x^4}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs.
\(2(123)=246\).
time = 0.12, size = 306, normalized size = 2.05
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-3 a d \,x^{2}-3 c \,x^{2} b +2 a c \right )}{8 a^{2} c^{2} x^{4}}+\frac {\left (-\frac {3 \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d^{2}}{16 c^{2} \sqrt {a c}}-\frac {\ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b d}{8 a c \sqrt {a c}}-\frac {3 \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b^{2}}{16 a^{2} \sqrt {a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(281\) |
default | \(-\frac {\left (3 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} d^{2} x^{4}+2 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a b c d \,x^{4}+3 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) b^{2} c^{2} x^{4}-6 d \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{2} a \sqrt {a c}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b c \,x^{2} \sqrt {a c}+4 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a c \sqrt {a c}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{16 a^{2} c^{2} \sqrt {a c}\, x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\) | \(306\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{4 a c \,x^{4}}+\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, d}{8 a \,c^{2} x^{2}}+\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, b}{8 a^{2} c \,x^{2}}-\frac {3 \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d^{2}}{16 c^{2} \sqrt {a c}}-\frac {\ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b d}{8 a c \sqrt {a c}}-\frac {3 \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b^{2}}{16 a^{2} \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.57, size = 360, normalized size = 2.42 \begin {gather*} \left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {a c}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, a^{3} c^{3} x^{4}}, \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-a c}}{2 \, {\left (a b c d x^{4} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, a^{3} c^{3} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1015 vs.
\(2 (123) = 246\).
time = 1.06, size = 1015, normalized size = 6.81 \begin {gather*} -\frac {\sqrt {b d} b^{6} d^{2} {\left (\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{2} b^{5} c^{2} d^{2}} - \frac {2 \, {\left (3 \, b^{8} c^{5} - 9 \, a b^{7} c^{4} d + 6 \, a^{2} b^{6} c^{3} d^{2} + 6 \, a^{3} b^{5} c^{2} d^{3} - 9 \, a^{4} b^{4} c d^{4} + 3 \, a^{5} b^{3} d^{5} - 9 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{6} c^{4} - 4 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{3} d + 26 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c^{2} d^{2} - 4 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c d^{3} - 9 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} d^{4} + 9 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{4} c^{3} + 15 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{2} d + 15 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c d^{2} + 9 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{3} b d^{3} - 3 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{2} c^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a b c d - 3 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a^{2} d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a^{2} b^{4} c^{2} d^{2}}\right )}}{8 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.81, size = 962, normalized size = 6.46 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {b\,x^2+a}-\sqrt {a}\,\sqrt {d\,x^2+c}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{16\,a^3\,c^3}-\frac {\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{16\,a^3\,c^3}-\frac {\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2\,\left (\frac {11\,a^2\,b^2\,d^2}{64}+\frac {5\,a\,b^3\,c\,d}{16}+\frac {11\,b^4\,c^2}{64}\right )}{a^{5/2}\,c^{5/2}\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}-\frac {b^4}{64\,a^{3/2}\,c^{3/2}\,d^2}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{32}-\frac {9\,a^2\,b^2\,c\,d^2}{16}-\frac {9\,a\,b^3\,c^2\,d}{16}+\frac {b^4\,c^3}{32}\right )}{a^3\,c^3\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}-\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {c\,b^4}{16}+\frac {a\,d\,b^3}{16}\right )}{a^2\,c^2\,d^2\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (\frac {a^3\,d^3}{8}-\frac {7\,a^2\,b\,c\,d^2}{32}-\frac {7\,a\,b^2\,c^2\,d}{32}+\frac {b^3\,c^3}{8}\right )}{a^3\,c^3\,d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4\,\left (-\frac {7\,a^4\,d^4}{64}+\frac {a^3\,b\,c\,d^3}{8}+\frac {45\,a^2\,b^2\,c^2\,d^2}{64}+\frac {a\,b^3\,c^3\,d}{8}-\frac {7\,b^4\,c^4}{64}\right )}{a^{7/2}\,c^{7/2}\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}+\frac {b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (2\,c\,b^2+2\,a\,d\,b\right )}{\sqrt {a}\,\sqrt {c}\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (2\,a\,d+2\,b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}}+\frac {d^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{64\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}+\frac {3\,d\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (a\,d+b\,c\right )}{32\,a^2\,c^2\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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