Optimal. Leaf size=74 \[ \frac {\sqrt {c+d x^4}}{2 b d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}} \]
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Rubi [A]
time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {457, 81, 65,
214} \begin {gather*} \frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^4}}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=\frac {\sqrt {c+d x^4}}{2 b d}-\frac {a \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{4 b}\\ &=\frac {\sqrt {c+d x^4}}{2 b d}-\frac {a \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{2 b d}\\ &=\frac {\sqrt {c+d x^4}}{2 b d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 73, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt {c+d x^4}}{b d}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{b^{3/2} \sqrt {-b c+a d}}\right ) \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. \(339\) vs.
\(2(58)=116\).
time = 0.36, size = 340, normalized size = 4.59
method | result | size |
risch | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(335\) |
elliptic | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(335\) |
default | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}-\frac {a \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b}\) | \(340\) |
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 = 2.64, size = 205, normalized size = 2.77 \begin {gather*} \left [\frac {\sqrt {b^{2} c - a b d} a d \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{4 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}, -\frac {\sqrt {-b^{2} c + a b d} a d \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{2 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}\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^{7}}{\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 [A]
time = 1.87, size = 64, normalized size = 0.86 \begin {gather*} -\frac {\frac {a d \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b} - \frac {\sqrt {d x^{4} + c}}{b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.73, size = 58, normalized size = 0.78 \begin {gather*} \frac {\sqrt {d\,x^4+c}}{2\,b\,d}-\frac {a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,b^{3/2}\,\sqrt {a\,d-b\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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