Optimal. Leaf size=213 \[ \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {(b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4} \]
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Rubi [A]
time = 0.22, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {382, 101, 156,
162, 65, 214, 211} \begin {gather*} \frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {(b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4}+\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{4 c^3 \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 101
Rule 156
Rule 162
Rule 211
Rule 214
Rule 382
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx &=-\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (b c-6 a d)-\frac {5 b d x}{2}}{x \sqrt {a+b x} (c+d x)^3} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\text {Subst}\left (\int \frac {-(b c-6 a d) (b c-a d)+\frac {9}{2} b d (b c-a d) x}{x \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{2 c^2 (b c-a d)}\\ &=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\text {Subst}\left (\int \frac {(b c-6 a d) (b c-a d)^2-\frac {1}{4} b d (11 b c-12 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3 (b c-a d)^2}\\ &=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {(b c-6 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{8 c^4 (b c-a d)}\\ &=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {(b c-6 a d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{4 b c^4 (b c-a d)}\\ &=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {(b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 189, normalized size = 0.89 \begin {gather*} \frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-2 a d \left (6 d^2+9 c d x+2 c^2 x^2\right )+b c \left (11 d^2+17 c d x+4 c^2 x^2\right )\right )}{(b c-a d) (d+c x)^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {4 (b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{4 c^4} \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. \(1971\) vs.
\(2(185)=370\).
time = 0.09, size = 1972, normalized size = 9.26
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1949\) |
default | \(\text {Expression too large to display}\) | \(1972\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs.
\(2 (185) = 370\).
time = 3.46, size = 1749, normalized size = 8.21 \begin {gather*} \left [-\frac {4 \, {\left (b^{2} c^{2} d^{2} - 7 \, a b c d^{3} + 6 \, a^{2} d^{4} + {\left (b^{2} c^{4} - 7 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 6 \, a^{2} c d^{3}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (15 \, a b^{2} c^{2} d^{2} - 40 \, a^{2} b c d^{3} + 24 \, a^{3} d^{4} + {\left (15 \, a b^{2} c^{4} - 40 \, a^{2} b c^{3} d + 24 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a b^{2} c^{3} d - 40 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3}\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (4 \, {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{3} + {\left (17 \, a b c^{3} d - 18 \, a^{2} c^{2} d^{2}\right )} x^{2} + {\left (11 \, a b c^{2} d^{2} - 12 \, a^{2} c d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, {\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x^{2} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x\right )}}, \frac {{\left (15 \, a b^{2} c^{2} d^{2} - 40 \, a^{2} b c d^{3} + 24 \, a^{3} d^{4} + {\left (15 \, a b^{2} c^{4} - 40 \, a^{2} b c^{3} d + 24 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a b^{2} c^{3} d - 40 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3}\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - 2 \, {\left (b^{2} c^{2} d^{2} - 7 \, a b c d^{3} + 6 \, a^{2} d^{4} + {\left (b^{2} c^{4} - 7 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 6 \, a^{2} c d^{3}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (4 \, {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{3} + {\left (17 \, a b c^{3} d - 18 \, a^{2} c^{2} d^{2}\right )} x^{2} + {\left (11 \, a b c^{2} d^{2} - 12 \, a^{2} c d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, {\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x^{2} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x\right )}}, -\frac {8 \, {\left (b^{2} c^{2} d^{2} - 7 \, a b c d^{3} + 6 \, a^{2} d^{4} + {\left (b^{2} c^{4} - 7 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 6 \, a^{2} c d^{3}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (15 \, a b^{2} c^{2} d^{2} - 40 \, a^{2} b c d^{3} + 24 \, a^{3} d^{4} + {\left (15 \, a b^{2} c^{4} - 40 \, a^{2} b c^{3} d + 24 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a b^{2} c^{3} d - 40 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3}\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (4 \, {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{3} + {\left (17 \, a b c^{3} d - 18 \, a^{2} c^{2} d^{2}\right )} x^{2} + {\left (11 \, a b c^{2} d^{2} - 12 \, a^{2} c d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, {\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x^{2} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x\right )}}, \frac {{\left (15 \, a b^{2} c^{2} d^{2} - 40 \, a^{2} b c d^{3} + 24 \, a^{3} d^{4} + {\left (15 \, a b^{2} c^{4} - 40 \, a^{2} b c^{3} d + 24 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a b^{2} c^{3} d - 40 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3}\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - 4 \, {\left (b^{2} c^{2} d^{2} - 7 \, a b c d^{3} + 6 \, a^{2} d^{4} + {\left (b^{2} c^{4} - 7 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 6 \, a^{2} c d^{3}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (4 \, {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{3} + {\left (17 \, a b c^{3} d - 18 \, a^{2} c^{2} d^{2}\right )} x^{2} + {\left (11 \, a b c^{2} d^{2} - 12 \, a^{2} c d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, {\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x^{2} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x\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^{3} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{3}}\, 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. 820 vs.
\(2 (185) = 370\).
time = 2.00, size = 820, normalized size = 3.85 \begin {gather*} -\frac {{\left (15 \, \sqrt {a} b^{2} c^{2} d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 40 \, a^{\frac {3}{2}} b c d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 24 \, a^{\frac {5}{2}} d^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 2 \, \sqrt {b c d - a d^{2}} b^{2} c^{2} \log \left ({\left | b \right |}\right ) + 14 \, \sqrt {b c d - a d^{2}} a b c d \log \left ({\left | b \right |}\right ) - 12 \, \sqrt {b c d - a d^{2}} a^{2} d^{2} \log \left ({\left | b \right |}\right ) + 9 \, \sqrt {b c d - a d^{2}} a b c d - 10 \, \sqrt {b c d - a d^{2}} a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{4 \, {\left (\sqrt {b c d - a d^{2}} \sqrt {a} b c^{5} - \sqrt {b c d - a d^{2}} a^{\frac {3}{2}} c^{4} d\right )}} - \frac {{\left (15 \, b^{2} c^{2} d \mathrm {sgn}\left (x\right ) - 40 \, a b c d^{2} \mathrm {sgn}\left (x\right ) + 24 \, a^{2} d^{3} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{4 \, {\left (b c^{5} - a c^{4} d\right )} \sqrt {b c d - a d^{2}}} + \frac {\sqrt {a x^{2} + b x} \mathrm {sgn}\left (x\right )}{c^{3}} - \frac {9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} \sqrt {a} b^{2} c^{3} d \mathrm {sgn}\left (x\right ) - 32 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b c^{2} d^{2} \mathrm {sgn}\left (x\right ) + 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {5}{2}} c d^{3} \mathrm {sgn}\left (x\right ) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{3} d^{4} \mathrm {sgn}\left (x\right ) + 7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 44 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {5}{2}} b d^{4} \mathrm {sgn}\left (x\right ) - 9 \, a b^{3} c d^{3} \mathrm {sgn}\left (x\right ) + 10 \, a^{2} b^{2} d^{4} \mathrm {sgn}\left (x\right )}{4 \, {\left (\sqrt {a} b c^{5} - a^{\frac {3}{2}} c^{4} d\right )} {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )}^{2}} - \frac {{\left (b c \mathrm {sgn}\left (x\right ) - 6 \, a d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, \sqrt {a} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.73, size = 1895, normalized size = 8.90 \begin {gather*} \frac {\ln \left (\sqrt {a+\frac {b}{x}}\,\sqrt {d\,{\left (a\,d-b\,c\right )}^3}-a^2\,d^2-b^2\,c^2+2\,a\,b\,c\,d\right )\,\sqrt {d\,{\left (a\,d-b\,c\right )}^3}\,\left (3\,a^2\,d^2-5\,a\,b\,c\,d+\frac {15\,b^2\,c^2}{8}\right )}{-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7}-\frac {\frac {b\,\sqrt {a+\frac {b}{x}}\,\left (12\,a^2\,d^2-17\,a\,b\,c\,d+4\,b^2\,c^2\right )}{4\,c^3}+\frac {b\,{\left (a+\frac {b}{x}\right )}^{5/2}\,\left (12\,a\,d^3-11\,b\,c\,d^2\right )}{4\,c^3\,\left (a\,d-b\,c\right )}-\frac {d\,{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (24\,a^2\,b\,d^2-40\,a\,b^2\,c\,d+17\,b^3\,c^2\right )}{4\,c^3\,\left (a\,d-b\,c\right )}}{{\left (a+\frac {b}{x}\right )}^2\,\left (3\,a\,d^2-2\,b\,c\,d\right )-\left (a+\frac {b}{x}\right )\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )-d^2\,{\left (a+\frac {b}{x}\right )}^3+a^3\,d^2+a\,b^2\,c^2-2\,a^2\,b\,c\,d}-\frac {\ln \left (\sqrt {a+\frac {b}{x}}\,\sqrt {d\,{\left (a\,d-b\,c\right )}^3}+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d\right )\,\sqrt {d\,{\left (a\,d-b\,c\right )}^3}\,\left (24\,a^2\,d^2-40\,a\,b\,c\,d+15\,b^2\,c^2\right )}{8\,\left (-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7\right )}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {a+\frac {b}{x}}\,\left (1152\,a^4\,b^2\,d^7-3264\,a^3\,b^3\,c\,d^6+3296\,a^2\,b^4\,c^2\,d^5-1424\,a\,b^5\,c^3\,d^4+241\,b^6\,c^4\,d^3\right )}{8\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}-\frac {\left (6\,a\,d-b\,c\right )\,\left (\frac {-12\,a^3\,b^3\,c^8\,d^5+29\,a^2\,b^4\,c^9\,d^4-21\,a\,b^5\,c^{10}\,d^3+4\,b^6\,c^{11}\,d^2}{a^2\,c^9\,d^2-2\,a\,b\,c^{10}\,d+b^2\,c^{11}}-\frac {\sqrt {a+\frac {b}{x}}\,\left (6\,a\,d-b\,c\right )\,\left (-128\,a^3\,b^2\,c^8\,d^5+320\,a^2\,b^3\,c^9\,d^4-256\,a\,b^4\,c^{10}\,d^3+64\,b^5\,c^{11}\,d^2\right )}{16\,\sqrt {a}\,c^4\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}\right )}{2\,\sqrt {a}\,c^4}\right )\,\left (6\,a\,d-b\,c\right )\,1{}\mathrm {i}}{2\,\sqrt {a}\,c^4}+\frac {\left (\frac {\sqrt {a+\frac {b}{x}}\,\left (1152\,a^4\,b^2\,d^7-3264\,a^3\,b^3\,c\,d^6+3296\,a^2\,b^4\,c^2\,d^5-1424\,a\,b^5\,c^3\,d^4+241\,b^6\,c^4\,d^3\right )}{8\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}+\frac {\left (6\,a\,d-b\,c\right )\,\left (\frac {-12\,a^3\,b^3\,c^8\,d^5+29\,a^2\,b^4\,c^9\,d^4-21\,a\,b^5\,c^{10}\,d^3+4\,b^6\,c^{11}\,d^2}{a^2\,c^9\,d^2-2\,a\,b\,c^{10}\,d+b^2\,c^{11}}+\frac {\sqrt {a+\frac {b}{x}}\,\left (6\,a\,d-b\,c\right )\,\left (-128\,a^3\,b^2\,c^8\,d^5+320\,a^2\,b^3\,c^9\,d^4-256\,a\,b^4\,c^{10}\,d^3+64\,b^5\,c^{11}\,d^2\right )}{16\,\sqrt {a}\,c^4\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}\right )}{2\,\sqrt {a}\,c^4}\right )\,\left (6\,a\,d-b\,c\right )\,1{}\mathrm {i}}{2\,\sqrt {a}\,c^4}}{\frac {216\,a^4\,b^3\,d^7-594\,a^3\,b^4\,c\,d^6+558\,a^2\,b^5\,c^2\,d^5-\frac {805\,a\,b^6\,c^3\,d^4}{4}+\frac {165\,b^7\,c^4\,d^3}{8}}{a^2\,c^9\,d^2-2\,a\,b\,c^{10}\,d+b^2\,c^{11}}-\frac {\left (\frac {\sqrt {a+\frac {b}{x}}\,\left (1152\,a^4\,b^2\,d^7-3264\,a^3\,b^3\,c\,d^6+3296\,a^2\,b^4\,c^2\,d^5-1424\,a\,b^5\,c^3\,d^4+241\,b^6\,c^4\,d^3\right )}{8\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}-\frac {\left (6\,a\,d-b\,c\right )\,\left (\frac {-12\,a^3\,b^3\,c^8\,d^5+29\,a^2\,b^4\,c^9\,d^4-21\,a\,b^5\,c^{10}\,d^3+4\,b^6\,c^{11}\,d^2}{a^2\,c^9\,d^2-2\,a\,b\,c^{10}\,d+b^2\,c^{11}}-\frac {\sqrt {a+\frac {b}{x}}\,\left (6\,a\,d-b\,c\right )\,\left (-128\,a^3\,b^2\,c^8\,d^5+320\,a^2\,b^3\,c^9\,d^4-256\,a\,b^4\,c^{10}\,d^3+64\,b^5\,c^{11}\,d^2\right )}{16\,\sqrt {a}\,c^4\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}\right )}{2\,\sqrt {a}\,c^4}\right )\,\left (6\,a\,d-b\,c\right )}{2\,\sqrt {a}\,c^4}+\frac {\left (\frac {\sqrt {a+\frac {b}{x}}\,\left (1152\,a^4\,b^2\,d^7-3264\,a^3\,b^3\,c\,d^6+3296\,a^2\,b^4\,c^2\,d^5-1424\,a\,b^5\,c^3\,d^4+241\,b^6\,c^4\,d^3\right )}{8\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}+\frac {\left (6\,a\,d-b\,c\right )\,\left (\frac {-12\,a^3\,b^3\,c^8\,d^5+29\,a^2\,b^4\,c^9\,d^4-21\,a\,b^5\,c^{10}\,d^3+4\,b^6\,c^{11}\,d^2}{a^2\,c^9\,d^2-2\,a\,b\,c^{10}\,d+b^2\,c^{11}}+\frac {\sqrt {a+\frac {b}{x}}\,\left (6\,a\,d-b\,c\right )\,\left (-128\,a^3\,b^2\,c^8\,d^5+320\,a^2\,b^3\,c^9\,d^4-256\,a\,b^4\,c^{10}\,d^3+64\,b^5\,c^{11}\,d^2\right )}{16\,\sqrt {a}\,c^4\,\left (a^2\,c^6\,d^2-2\,a\,b\,c^7\,d+b^2\,c^8\right )}\right )}{2\,\sqrt {a}\,c^4}\right )\,\left (6\,a\,d-b\,c\right )}{2\,\sqrt {a}\,c^4}}\right )\,\left (6\,a\,d-b\,c\right )\,1{}\mathrm {i}}{\sqrt {a}\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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