Optimal. Leaf size=175 \[ \frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}-\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 100, 152,
65, 214} \begin {gather*} -\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}+\frac {a x^8 \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 65
Rule 100
Rule 152
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\text {Subst}\left (\int \frac {x \left (2 a c+\frac {1}{2} (-2 b c+5 a d) x\right )}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{4 b (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac {\left (a^2 (6 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{8 b^3 (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac {\left (a^2 (6 b c-5 a d)\right ) \text {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^3 d (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}-\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 175, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c+d x^4} \left (-15 a^3 d^2+2 a^2 b d \left (4 c-5 d x^4\right )+2 b^3 c x^4 \left (2 c-d x^4\right )+2 a b^2 \left (2 c^2+3 c d x^4+d^2 x^8\right )\right )}{12 b^3 d^2 (b c-a d) \left (a+b x^4\right )}+\frac {a^2 (-6 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{4 b^{7/2} (-b c+a 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. \(917\) vs.
\(2(155)=310\).
time = 0.39, size = 918, normalized size = 5.25 Too large to display
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 = 3.17, size = 622, normalized size = 3.55 \begin {gather*} \left [\frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\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 \, {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}, \frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\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 ) + {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 2.22, size = 180, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d x^{4} + c} a^{3} d}{4 \, {\left (b^{4} c - a b^{3} d\right )} {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac {{\left (6 \, a^{2} b c - 5 \, a^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{4} d^{4} - 3 \, \sqrt {d x^{4} + c} b^{4} c d^{4} - 6 \, \sqrt {d x^{4} + c} a b^{3} d^{5}}{6 \, b^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.19, size = 186, normalized size = 1.06 \begin {gather*} \frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b^2\,d^2}-\left (\frac {3\,c}{2\,b^2\,d^2}+\frac {a\,d-b\,c}{b^3\,d^2}\right )\,\sqrt {d\,x^4+c}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\left (5\,a\,d-6\,b\,c\right )}{\sqrt {a\,d-b\,c}\,\left (5\,a^3\,d-6\,a^2\,b\,c\right )}\right )\,\left (5\,a\,d-6\,b\,c\right )}{4\,b^{7/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a^3\,d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b^4\,\left (d\,x^4+c\right )-2\,b^4\,c+2\,a\,b^3\,d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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