Optimal. Leaf size=207 \[ \frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {2 \sqrt {b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4} \]
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Rubi [A]
time = 0.19, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 154, 156,
162, 65, 214} \begin {gather*} -\frac {2 \sqrt {b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4}+\frac {c \sqrt {c+d x} (2 b c-3 a d)}{4 a^2 x^2}-\frac {\sqrt {c+d x} \left (11 a^2 d^2-18 a b c d+8 b^2 c^2\right )}{8 a^3 x}+\frac {\left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {c (c+d x)^{3/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 154
Rule 156
Rule 162
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)} \, dx &=-\frac {c (c+d x)^{3/2}}{3 a x^3}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{2} c (2 b c-3 a d)+\frac {3}{2} d (b c-2 a d) x\right )}{x^3 (a+b x)} \, dx}{3 a}\\ &=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {c (c+d x)^{3/2}}{3 a x^3}-\frac {\int \frac {-\frac {3}{4} c \left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right )-\frac {3}{4} d \left (6 b^2 c^2-13 a b c d+8 a^2 d^2\right ) x}{x^2 (a+b x) \sqrt {c+d x}} \, dx}{6 a^2}\\ &=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\int \frac {-\frac {3}{8} c \left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right )-\frac {3}{8} b c d \left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{6 a^3 c}\\ &=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (b (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{a^4}-\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx}{16 a^4}\\ &=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (2 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^4 d}-\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 a^4 d}\\ &=\frac {c (2 b c-3 a d) \sqrt {c+d x}}{4 a^2 x^2}-\frac {\left (8 b^2 c^2-18 a b c d+11 a^2 d^2\right ) \sqrt {c+d x}}{8 a^3 x}-\frac {c (c+d x)^{3/2}}{3 a x^3}+\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^4 \sqrt {c}}-\frac {2 \sqrt {b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 178, normalized size = 0.86 \begin {gather*} -\frac {\frac {a \sqrt {c+d x} \left (24 b^2 c^2 x^2-6 a b c x (2 c+9 d x)+a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{x^3}+48 \sqrt {b} (-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )-\frac {3 \left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{24 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 250, normalized size = 1.21
method | result | size |
derivativedivides | \(2 d^{4} \left (-\frac {b \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4} \sqrt {\left (a d -b c \right ) b}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{8} a^{2} b c \,d^{2}+\frac {1}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+2 a^{2} b \,c^{2} d^{2}-a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} a^{2} b \,c^{3} d^{2}+\frac {1}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{4} d^{4}}\right )\) | \(250\) |
default | \(2 d^{4} \left (-\frac {b \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{4} d^{4} \sqrt {\left (a d -b c \right ) b}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{8} a^{2} b c \,d^{2}+\frac {1}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+2 a^{2} b \,c^{2} d^{2}-a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{8} a^{2} b \,c^{3} d^{2}+\frac {1}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{4} d^{4}}\right )\) | \(250\) |
risch | \(-\frac {\sqrt {d x +c}\, \left (33 a^{2} d^{2} x^{2}-54 a b c d \,x^{2}+24 b^{2} c^{2} x^{2}+26 a^{2} c d x -12 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{3} x^{3}}-\frac {2 d^{3} b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a \sqrt {\left (a d -b c \right ) b}}+\frac {6 d^{2} b^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c}{a^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {6 d \,b^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{2}}{a^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {2 b^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{3}}{a^{4} \sqrt {\left (a d -b c \right ) b}}-\frac {5 d^{3} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 a \sqrt {c}}+\frac {15 d^{2} \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b}{4 a^{2}}-\frac {5 d \,c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b^{2}}{a^{3}}+\frac {2 c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b^{3}}{a^{4}}\) | \(347\) |
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 = 1.73, size = 930, normalized size = 4.49 \begin {gather*} \left [\frac {48 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {b^{2} c - a b d} x^{3} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {c} x^{3} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, a^{4} c x^{3}}, \frac {96 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c + a b d} x^{3} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {c} x^{3} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, a^{4} c x^{3}}, -\frac {3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - 24 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {b^{2} c - a b d} x^{3} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, a^{4} c x^{3}}, \frac {48 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c + a b d} x^{3} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - {\left (8 \, a^{3} c^{3} + 3 \, {\left (8 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d + 11 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (6 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, a^{4} c x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1273 vs.
\(2 (199) = 398\).
time = 101.97, size = 1273, normalized size = 6.15 \begin {gather*} \frac {10 b c^{4} d^{2} \sqrt {c + d x}}{- 8 a^{2} c^{4} - 16 a^{2} c^{3} d x + 8 a^{2} c^{2} \left (c + d x\right )^{2}} - \frac {6 b c^{3} d^{2} \left (c + d x\right )^{\frac {3}{2}}}{- 8 a^{2} c^{4} - 16 a^{2} c^{3} d x + 8 a^{2} c^{2} \left (c + d x\right )^{2}} - \frac {66 c^{5} d^{3} \sqrt {c + d x}}{96 a c^{6} + 144 a c^{5} d x - 144 a c^{4} \left (c + d x\right )^{2} + 48 a c^{3} \left (c + d x\right )^{3}} + \frac {80 c^{4} d^{3} \left (c + d x\right )^{\frac {3}{2}}}{96 a c^{6} + 144 a c^{5} d x - 144 a c^{4} \left (c + d x\right )^{2} + 48 a c^{3} \left (c + d x\right )^{3}} - \frac {30 c^{3} d^{3} \left (c + d x\right )^{\frac {5}{2}}}{96 a c^{6} + 144 a c^{5} d x - 144 a c^{4} \left (c + d x\right )^{2} + 48 a c^{3} \left (c + d x\right )^{3}} - \frac {30 c^{3} d^{3} \sqrt {c + d x}}{- 8 a c^{4} - 16 a c^{3} d x + 8 a c^{2} \left (c + d x\right )^{2}} + \frac {18 c^{2} d^{3} \left (c + d x\right )^{\frac {3}{2}}}{- 8 a c^{4} - 16 a c^{3} d x + 8 a c^{2} \left (c + d x\right )^{2}} - \frac {5 c^{3} d^{3} \sqrt {\frac {1}{c^{7}}} \log {\left (- c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {c + d x} \right )}}{16 a} + \frac {5 c^{3} d^{3} \sqrt {\frac {1}{c^{7}}} \log {\left (c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {c + d x} \right )}}{16 a} + \frac {9 c^{2} d^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a} - \frac {9 c^{2} d^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a} - \frac {3 c d^{3} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a} + \frac {3 c d^{3} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {2 d^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a \sqrt {\frac {a d}{b} - c}} + \frac {2 d^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {3 d^{2} \sqrt {c + d x}}{a x} - \frac {3 b c^{3} d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a^{2}} + \frac {3 b c^{3} d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a^{2}} + \frac {3 b c^{2} d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {3 b c^{2} d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {6 b c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{2} \sqrt {\frac {a d}{b} - c}} - \frac {6 b c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} + \frac {3 b c d \sqrt {c + d x}}{a^{2} x} - \frac {b^{2} c^{3} d \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{3}} + \frac {b^{2} c^{3} d \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{3}} - \frac {6 b^{2} c^{2} d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{3} \sqrt {\frac {a d}{b} - c}} + \frac {6 b^{2} c^{2} d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{3} \sqrt {- c}} - \frac {b^{2} c^{2} \sqrt {c + d x}}{a^{3} x} + \frac {2 b^{3} c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{4} \sqrt {\frac {a d}{b} - c}} - \frac {2 b^{3} c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{4} \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 300, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{4}} - \frac {{\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{8 \, a^{4} \sqrt {-c}} - \frac {24 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 48 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 24 \, \sqrt {d x + c} b^{2} c^{4} d - 54 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{2} + 96 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 42 \, \sqrt {d x + c} a b c^{3} d^{2} + 33 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 15 \, \sqrt {d x + c} a^{2} c^{2} d^{3}}{24 \, a^{3} d^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 2147, normalized size = 10.37 \begin {gather*} \frac {\frac {\sqrt {c+d\,x}\,\left (5\,a^2\,c^2\,d^3-14\,a\,b\,c^3\,d^2+8\,b^2\,c^4\,d\right )}{8\,a^3}-\frac {{\left (c+d\,x\right )}^{3/2}\,\left (5\,a^2\,c\,d^3-12\,a\,b\,c^2\,d^2+6\,b^2\,c^3\,d\right )}{3\,a^3}+\frac {d\,{\left (c+d\,x\right )}^{5/2}\,\left (11\,a^2\,d^2-18\,a\,b\,c\,d+8\,b^2\,c^2\right )}{8\,a^3}}{3\,c\,{\left (c+d\,x\right )}^2-3\,c^2\,\left (c+d\,x\right )-{\left (c+d\,x\right )}^3+c^3}+\frac {2\,\mathrm {atanh}\left (\frac {25\,b^3\,d^8\,\sqrt {c+d\,x}\,\sqrt {-a^5\,b\,d^5+5\,a^4\,b^2\,c\,d^4-10\,a^3\,b^3\,c^2\,d^3+10\,a^2\,b^4\,c^3\,d^2-5\,a\,b^5\,c^4\,d+b^6\,c^5}}{16\,\left (\frac {25\,a^3\,b^3\,d^{11}}{16}-\frac {217\,b^6\,c^3\,d^8}{16}+\frac {227\,a\,b^5\,c^2\,d^9}{16}-\frac {119\,a^2\,b^4\,c\,d^{10}}{16}+\frac {13\,b^7\,c^4\,d^7}{2\,a}-\frac {5\,b^8\,c^5\,d^6}{4\,a^2}\right )}+\frac {5\,b^5\,c^2\,d^6\,\sqrt {c+d\,x}\,\sqrt {-a^5\,b\,d^5+5\,a^4\,b^2\,c\,d^4-10\,a^3\,b^3\,c^2\,d^3+10\,a^2\,b^4\,c^3\,d^2-5\,a\,b^5\,c^4\,d+b^6\,c^5}}{4\,\left (\frac {25\,a^5\,b^3\,d^{11}}{16}-\frac {119\,a^4\,b^4\,c\,d^{10}}{16}+\frac {227\,a^3\,b^5\,c^2\,d^9}{16}-\frac {217\,a^2\,b^6\,c^3\,d^8}{16}+\frac {13\,a\,b^7\,c^4\,d^7}{2}-\frac {5\,b^8\,c^5\,d^6}{4}\right )}-\frac {11\,b^4\,c\,d^7\,\sqrt {c+d\,x}\,\sqrt {-a^5\,b\,d^5+5\,a^4\,b^2\,c\,d^4-10\,a^3\,b^3\,c^2\,d^3+10\,a^2\,b^4\,c^3\,d^2-5\,a\,b^5\,c^4\,d+b^6\,c^5}}{4\,\left (\frac {25\,a^4\,b^3\,d^{11}}{16}+\frac {13\,b^7\,c^4\,d^7}{2}-\frac {217\,a\,b^6\,c^3\,d^8}{16}-\frac {119\,a^3\,b^4\,c\,d^{10}}{16}+\frac {227\,a^2\,b^5\,c^2\,d^9}{16}-\frac {5\,b^8\,c^5\,d^6}{4\,a}\right )}\right )\,\sqrt {-b\,{\left (a\,d-b\,c\right )}^5}}{a^4}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {c+d\,x}\,\left (281\,a^6\,b^3\,d^8-1836\,a^5\,b^4\,c\,d^7+5140\,a^4\,b^5\,c^2\,d^6-7680\,a^3\,b^6\,c^3\,d^5+6400\,a^2\,b^7\,c^4\,d^4-2816\,a\,b^8\,c^5\,d^3+512\,b^9\,c^6\,d^2\right )}{32\,a^6}-\frac {\left (\frac {80\,a^{11}\,b^2\,d^6-304\,a^{10}\,b^3\,c\,d^5+352\,a^9\,b^4\,c^2\,d^4-128\,a^8\,b^5\,c^3\,d^3}{32\,a^9}-\frac {\left (256\,a^9\,b^2\,d^3-512\,a^8\,b^3\,c\,d^2\right )\,\sqrt {c+d\,x}\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{512\,a^{10}\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )\,1{}\mathrm {i}}{16\,a^4\,\sqrt {c}}+\frac {\left (\frac {\sqrt {c+d\,x}\,\left (281\,a^6\,b^3\,d^8-1836\,a^5\,b^4\,c\,d^7+5140\,a^4\,b^5\,c^2\,d^6-7680\,a^3\,b^6\,c^3\,d^5+6400\,a^2\,b^7\,c^4\,d^4-2816\,a\,b^8\,c^5\,d^3+512\,b^9\,c^6\,d^2\right )}{32\,a^6}+\frac {\left (\frac {80\,a^{11}\,b^2\,d^6-304\,a^{10}\,b^3\,c\,d^5+352\,a^9\,b^4\,c^2\,d^4-128\,a^8\,b^5\,c^3\,d^3}{32\,a^9}+\frac {\left (256\,a^9\,b^2\,d^3-512\,a^8\,b^3\,c\,d^2\right )\,\sqrt {c+d\,x}\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{512\,a^{10}\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )\,1{}\mathrm {i}}{16\,a^4\,\sqrt {c}}}{\frac {55\,a^8\,b^3\,d^{11}-585\,a^7\,b^4\,c\,d^{10}+2445\,a^6\,b^5\,c^2\,d^9-5511\,a^5\,b^6\,c^3\,d^8+7496\,a^4\,b^7\,c^4\,d^7-6380\,a^3\,b^8\,c^5\,d^6+3344\,a^2\,b^9\,c^6\,d^5-992\,a\,b^{10}\,c^7\,d^4+128\,b^{11}\,c^8\,d^3}{16\,a^9}-\frac {\left (\frac {\sqrt {c+d\,x}\,\left (281\,a^6\,b^3\,d^8-1836\,a^5\,b^4\,c\,d^7+5140\,a^4\,b^5\,c^2\,d^6-7680\,a^3\,b^6\,c^3\,d^5+6400\,a^2\,b^7\,c^4\,d^4-2816\,a\,b^8\,c^5\,d^3+512\,b^9\,c^6\,d^2\right )}{32\,a^6}-\frac {\left (\frac {80\,a^{11}\,b^2\,d^6-304\,a^{10}\,b^3\,c\,d^5+352\,a^9\,b^4\,c^2\,d^4-128\,a^8\,b^5\,c^3\,d^3}{32\,a^9}-\frac {\left (256\,a^9\,b^2\,d^3-512\,a^8\,b^3\,c\,d^2\right )\,\sqrt {c+d\,x}\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{512\,a^{10}\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}+\frac {\left (\frac {\sqrt {c+d\,x}\,\left (281\,a^6\,b^3\,d^8-1836\,a^5\,b^4\,c\,d^7+5140\,a^4\,b^5\,c^2\,d^6-7680\,a^3\,b^6\,c^3\,d^5+6400\,a^2\,b^7\,c^4\,d^4-2816\,a\,b^8\,c^5\,d^3+512\,b^9\,c^6\,d^2\right )}{32\,a^6}+\frac {\left (\frac {80\,a^{11}\,b^2\,d^6-304\,a^{10}\,b^3\,c\,d^5+352\,a^9\,b^4\,c^2\,d^4-128\,a^8\,b^5\,c^3\,d^3}{32\,a^9}+\frac {\left (256\,a^9\,b^2\,d^3-512\,a^8\,b^3\,c\,d^2\right )\,\sqrt {c+d\,x}\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{512\,a^{10}\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )}{16\,a^4\,\sqrt {c}}}\right )\,\left (5\,a^3\,d^3-30\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d-16\,b^3\,c^3\right )\,1{}\mathrm {i}}{8\,a^4\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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