Optimal. Leaf size=141 \[ -\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{5/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 141, 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, 92, 81, 65,
223, 212} \begin {gather*} -\frac {\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}-\frac {3 \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}+\frac {\text {Subst}\left (\int \frac {-a c-\frac {3}{2} (b c+a d) x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 b d}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 b^2 d^2}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{8 b^3 d^2}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{8 b^3 d^2}\\ &=-\frac {3 (b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2}+\frac {x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.29, size = 120, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-3 b c-3 a d+2 b d x^2\right )}{8 b^2 d^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{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. \(290\) vs.
\(2(115)=230\).
time = 0.11, size = 291, normalized size = 2.06
method | result | size |
risch | \(-\frac {\left (-2 b d \,x^{2}+3 a d +3 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{8 b^{2} d^{2}}+\frac {\left (\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2}}{16 b^{2} \sqrt {b d}}+\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a c}{8 b d \sqrt {b d}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{2}}{16 d^{2} \sqrt {b d}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(260\) |
default | \(\frac {\left (4 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b d \,x^{2}+3 d^{2} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2}+2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a c b d +3 b^{2} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) c^{2}-6 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, a d -6 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{16 \sqrt {b d}\, d^{2} b^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\) | \(291\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{4 b d}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{8 b^{2} d}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{8 b \,d^{2}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2}}{16 b^{2} \sqrt {b d}}+\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a c}{8 b d \sqrt {b d}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{2}}{16 d^{2} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(315\) |
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 = 1.11, size = 336, normalized size = 2.38 \begin {gather*} \left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, b^{3} d^{3}}, -\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, b^{3} d^{3}}\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 {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 [A]
time = 1.15, size = 160, normalized size = 1.13 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (\frac {2 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {3 \, b^{6} c d + 5 \, a b^{5} d^{2}}{b^{8} d^{3}}\right )} - \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\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 |}\right )}{\sqrt {b d} b^{2} d^{2}}\right )} b}{8 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.87, size = 550, normalized size = 3.90 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{4\,b^{5/2}\,d^{5/2}}-\frac {\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {3\,a^2\,b\,d^2}{4}+\frac {a\,b^2\,c\,d}{2}+\frac {3\,b^3\,c^2}{4}\right )}{d^6\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (\frac {11\,a^2\,d^2}{4}+\frac {25\,a\,b\,c\,d}{2}+\frac {11\,b^2\,c^2}{4}\right )}{d^5\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^7\,\left (\frac {3\,a^2\,d^2}{4}+\frac {a\,b\,c\,d}{2}+\frac {3\,b^2\,c^2}{4}\right )}{b^2\,d^3\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (\frac {11\,a^2\,d^2}{4}+\frac {25\,a\,b\,c\,d}{2}+\frac {11\,b^2\,c^2}{4}\right )}{b\,d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4\,\left (16\,a\,d+16\,b\,c\right )}{d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^8}+\frac {b^4}{d^4}-\frac {4\,b^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d^3\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}-\frac {4\,b\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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