Optimal. Leaf size=143 \[ \frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4} \]
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Rubi [A] time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {446, 96, 94, 93, 208} \[ \frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{x^5 \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4}-\frac {\left (\frac {b c}{2}+\frac {3 a d}{2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a c}\\ &=\frac {(b c+3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4}-\frac {((b c-a d) (b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 a c^2}\\ &=\frac {(b c+3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4}-\frac {((b c-a d) (b c+3 a d)) \operatorname {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 c^2}\\ &=\frac {(b c+3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 a c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 a c x^4}+\frac {(b c-a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 125, normalized size = 0.87 \[ \frac {\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a c+3 a d x^2-b c x^2\right )}{8 a c^2 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 358, normalized size = 2.50 \[ \left [-\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {a c}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, a^{2} c^{3} x^{4}}, -\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-a c}}{2 \, {\left (a b c d x^{4} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, a^{2} c^{3} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.27, size = 1107, normalized size = 7.74 \[ \frac {b {\left (\frac {{\left (\sqrt {b d} b^{3} c^{2} + 2 \, \sqrt {b d} a b^{2} c d - 3 \, \sqrt {b d} a^{2} b 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 b c^{2}} - \frac {2 \, {\left (\sqrt {b d} b^{9} c^{5} - 7 \, \sqrt {b d} a b^{8} c^{4} d + 18 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} - 22 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} + 13 \, \sqrt {b d} a^{4} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{5} b^{4} d^{5} - 3 \, \sqrt {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} b^{7} c^{4} + 16 \, \sqrt {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} a b^{6} c^{3} d - 14 \, \sqrt {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} a^{2} b^{5} c^{2} d^{2} - 8 \, \sqrt {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} a^{3} b^{4} c d^{3} + 9 \, \sqrt {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} a^{4} b^{3} d^{4} + 3 \, \sqrt {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} b^{5} c^{3} - 7 \, \sqrt {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} a b^{4} c^{2} d - 3 \, \sqrt {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} a^{2} b^{3} c d^{2} - 9 \, \sqrt {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} a^{3} b^{2} d^{3} - \sqrt {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 )}^{6} b^{3} c^{2} - 2 \, \sqrt {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 )}^{6} a b^{2} c d + 3 \, \sqrt {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 )}^{6} a^{2} b 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 c^{2}}\right )}}{8 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 355, normalized size = 2.48 \[ -\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 a^{2} d^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-2 a b c d \,x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-b^{2} c^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-6 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a d \,x^{2}+2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b c \,x^{2}+4 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a c \right )}{16 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,c^{2} x^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 21.52, size = 955, normalized size = 6.68 \[ \frac {\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )\,\left (\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^2\,c^3}-\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {b\,d}{8\,a\,c}-\frac {3\,d\,\left (a\,d+b\,c\right )}{32\,a\,c^2}\right )}{\sqrt {d\,x^2+c}-\sqrt {c}}-\frac {\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (\frac {a^2\,d^2}{8}-\frac {11\,a\,b\,c\,d}{32}+\frac {5\,b^2\,c^2}{32}\right )}{a\,c^3\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}-\frac {b^4}{64\,\sqrt {a}\,c^{3/2}\,d^2}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2\,\left (\frac {11\,a^2\,b^2\,d^2}{64}+\frac {a\,b^3\,c\,d}{16}-\frac {5\,b^4\,c^2}{64}\right )}{a^{3/2}\,c^{5/2}\,d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{32}-\frac {9\,a^2\,b^2\,c\,d^2}{16}+\frac {3\,a\,b^3\,c^2\,d}{16}+\frac {b^4\,c^3}{32}\right )}{a^2\,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 {b^4\,c}{16}-\frac {a\,b^3\,d}{16}\right )}{a\,c^2\,d^2\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}+\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}{4}+\frac {21\,a^2\,b^2\,c^2\,d^2}{64}-\frac {a\,b^3\,c^3\,d}{4}+\frac {b^4\,c^4}{64}\right )}{a^{5/2}\,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 {\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 (\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^2\,c^3}+\frac {d^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{64\,\sqrt {a}\,c^{3/2}\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2} \]
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 \[ \int \frac {\sqrt {a + b x^{2}}}{x^{5} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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