Optimal. Leaf size=99 \[ \frac {a \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 99, 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 = {446, 78, 63, 208} \begin {gather*} \frac {a \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=\frac {a \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(2 b c-a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{8 b (b c-a d)}\\ &=\frac {a \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(2 b c-a d) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{4 b d (b c-a d)}\\ &=\frac {a \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 98, normalized size = 0.99 \begin {gather*} \frac {\frac {a \sqrt {b} \sqrt {c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}+\frac {(a d-2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}}{4 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 109, normalized size = 1.10 \begin {gather*} \frac {(2 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4} \sqrt {a d-b c}}{b c-a d}\right )}{4 b^{3/2} (a d-b c)^{3/2}}+\frac {a \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 348, normalized size = 3.52 \begin {gather*} \left [\frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + 2 \, a b c - a^{2} d\right )} \sqrt {b^{2} c - a b 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 (a b^{2} c - a^{2} b d\right )}}{8 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{4}\right )}}, \frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + 2 \, a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b 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 (a b^{2} c - a^{2} b d\right )}}{4 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 116, normalized size = 1.17 \begin {gather*} \frac {\frac {\sqrt {d x^{4} + c} a d^{2}}{{\left (b^{2} c - a b d\right )} {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 851, normalized size = 8.60 \begin {gather*} \frac {a d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, b^{2}}+\frac {a d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, b^{2}}-\frac {\ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{2}}-\frac {\ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{2}}-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{8 \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) b^{2}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{8 \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) b^{2}} \end {gather*}
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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mupad [B] time = 4.93, size = 95, normalized size = 0.96 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )\,\left (a\,d-2\,b\,c\right )}{4\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a\,d\,\sqrt {d\,x^4+c}}{2\,b\,\left (a\,d-b\,c\right )\,\left (2\,b\,\left (d\,x^4+c\right )+2\,a\,d-2\,b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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