Optimal. Leaf size=123 \[ \frac {x^2 \sqrt {c+d x^4}}{4 b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 d^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {476, 490, 537,
223, 212, 385, 211} \begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b^2 \sqrt {b c-a d}}-\frac {(2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 d^{3/2}}+\frac {x^2 \sqrt {c+d x^4}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 476
Rule 490
Rule 537
Rubi steps
\begin {align*} \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )\\ &=\frac {x^2 \sqrt {c+d x^4}}{4 b d}-\frac {\text {Subst}\left (\int \frac {a c+(b c+2 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 b d}\\ &=\frac {x^2 \sqrt {c+d x^4}}{4 b d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{2 b^2}-\frac {(b c+2 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 b^2 d}\\ &=\frac {x^2 \sqrt {c+d x^4}}{4 b d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{2 b^2}-\frac {(b c+2 a d) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 d}\\ &=\frac {x^2 \sqrt {c+d x^4}}{4 b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 140, normalized size = 1.14 \begin {gather*} \frac {\sqrt {d} \left (b x^2 \sqrt {c+d x^4}+\frac {2 a^{3/2} d \tan ^{-1}\left (\frac {a \sqrt {d}+b \sqrt {d} x^4+b x^2 \sqrt {c+d x^4}}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}\right )-(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {d} x^2}\right )}{4 b^2 d^{3/2}} \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. \(402\) vs.
\(2(99)=198\).
time = 0.38, size = 403, normalized size = 3.28
method | result | size |
default | \(\frac {\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 d}-\frac {c \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{4 d^{\frac {3}{2}}}}{b}-\frac {a \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{2 b^{2} \sqrt {d}}+\frac {a^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}\) | \(403\) |
risch | \(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b d}-\frac {a \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{2 b^{2} \sqrt {d}}-\frac {\ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right ) c}{4 b \,d^{\frac {3}{2}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(408\) |
elliptic | \(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b d}-\frac {a \ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right )}{2 b^{2} \sqrt {d}}-\frac {\ln \left (x^{2} \sqrt {d}+\sqrt {d \,x^{4}+c}\right ) c}{4 b \,d^{\frac {3}{2}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.73, size = 739, normalized size = 6.01 \begin {gather*} \left [\frac {2 \, \sqrt {d x^{4} + c} b d x^{2} + a d^{2} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{8 \, b^{2} d^{2}}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} + a d^{2} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 2 \, {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right )}{8 \, b^{2} d^{2}}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} - 2 \, a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c x^{2}\right )}}\right ) + {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{8 \, b^{2} d^{2}}, \frac {\sqrt {d x^{4} + c} b d x^{2} - a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c x^{2}\right )}}\right ) + {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right )}{4 \, b^{2} d^{2}}\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^{9}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int \frac {x^9}{\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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