Optimal. Leaf size=191 \[ \frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}} \]
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Rubi [A]
time = 0.22, 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 = {476, 481, 596,
537, 223, 212, 385, 211} \begin {gather*} \frac {a^{3/2} (5 b c-4 a d) \text {ArcTan}\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 476
Rule 481
Rule 537
Rule 596
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \text {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 {\text {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 {\text {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 ) \text {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) \text {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 ) \text {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) \text {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 = 1.66, size = 188, normalized size = 0.98 \begin {gather*} \frac {\frac {b x^2 \sqrt {c+d x^4} \left (-2 a^2 d+b^2 c x^4+a b \left (c-d x^4\right )\right )}{d (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {(b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {d} x^2}\right )}{d^{3/2}}}{4 b^3} \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. \(1277\) vs.
\(2(163)=326\).
time = 0.41, size = 1278, normalized size = 6.69 Too large to display
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 = 4.15, size = 1386, normalized size = 7.26 \begin {gather*} \left [\frac {2 \, {\left (a b^{2} c^{2} + 3 \, a^{2} b c d - 4 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 3 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right ) + {\left (5 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} + {\left (5 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x^{4}\right )} \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 ) + 4 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} + {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{4} + c}}{16 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\right )}}, \frac {4 \, {\left (a b^{2} c^{2} + 3 \, a^{2} b c d - 4 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 3 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right ) + {\left (5 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} + {\left (5 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x^{4}\right )} \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 ) + 4 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} + {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{4} + c}}{16 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\right )}}, -\frac {{\left (5 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} + {\left (5 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x^{4}\right )} \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 (a b^{2} c^{2} + 3 \, a^{2} b c d - 4 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 3 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right ) - 2 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} + {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{4} + c}}{8 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\right )}}, \frac {2 \, {\left (a b^{2} c^{2} + 3 \, a^{2} b c d - 4 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 3 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right ) - {\left (5 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} + {\left (5 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x^{4}\right )} \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 ) + 2 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{6} + {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{4} + c}}{8 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4}\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^{13}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 337 vs.
\(2 (163) = 326\).
time = 1.78, size = 337, normalized size = 1.76 \begin {gather*} -\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}} \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^{13}}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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