Optimal. Leaf size=191 \[ \frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}}+\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.37, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {465, 470, 582, 523, 217, 206, 377, 205} \[ \frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}}+\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 377
Rule 465
Rule 470
Rule 523
Rule 582
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a c-2 (b c-2 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 b (b c-a d)}\\ &=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {-2 a c (b c-2 a d)-2 (b c-a d) (b c+4 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{8 b^2 d (b c-a d)}\\ &=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {\left (a^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 b^3 (b c-a d)}-\frac {(b c+4 a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 b^3 d}\\ &=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {\left (a^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)}-\frac {(b c+4 a d) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d}\\ &=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 150, normalized size = 0.79 \[ \frac {\frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{(b c-a d)^{3/2}}+b x^2 \sqrt {c+d x^4} \left (\frac {a^2}{\left (a+b x^4\right ) (a d-b c)}+\frac {1}{d}\right )-\frac {(4 a d+b c) \log \left (\sqrt {d} \sqrt {c+d x^4}+d x^2\right )}{d^{3/2}}}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.75, size = 1386, normalized size = 7.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 337, normalized size = 1.76 \[ -\frac {{\left (5 \, a^{2} b c \sqrt {d} - 4 \, a^{3} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {\sqrt {d x^{4} + c} x^{2}}{4 \, b^{2} d} + \frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{3} d^{\frac {3}{2}} - a^{2} b c^{2} \sqrt {d}}{2 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{4} c - a b^{3} d\right )}} + \frac {{\left (b c \sqrt {d} + 4 \, a d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2}\right )}{8 \, b^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 953, normalized size = 4.99 \[ -\frac {5 a^{2} \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 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, b^{3}}+\frac {5 a^{2} \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 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, b^{3}}-\frac {\sqrt {-a b}\, a^{2} 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^{4}}+\frac {\sqrt {-a b}\, a^{2} 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^{4}}+\frac {\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}}\, a^{2}}{8 \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) b^{3}}+\frac {\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}}\, a^{2}}{8 \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) b^{3}}+\frac {\sqrt {d \,x^{4}+c}\, x^{2}}{4 b^{2} d}-\frac {a \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{b^{3} \sqrt {d}}-\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{4 b^{2} d^{\frac {3}{2}}} \]
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 \[ \int \frac {x^{13}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}}\,{d x} \]
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 \[ \int \frac {x^{13}}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]
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 \[ \int \frac {x^{13}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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