Optimal. Leaf size=339 \[ \frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}-\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} \sqrt {c}}+\frac {5}{4} \sqrt {b} \sqrt {d} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {99, 154, 159,
163, 65, 223, 212, 95, 214} \begin {gather*} -\frac {5 \sqrt {a+b x} (c+d x)^{5/2} \left (a^2 d^2+12 a b c d+3 b^2 c^2\right )}{24 c^2 x}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2} \left (a^2 d^2+14 a b c d+9 b^2 c^2\right )}{24 c^2}+\frac {5 d \sqrt {a+b x} \sqrt {c+d x} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right )}{8 c}-\frac {5 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} \sqrt {c}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{12 c x^2}+\frac {5}{4} \sqrt {b} \sqrt {d} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^4} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {1}{3} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x^3} \, dx\\ &=-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {5}{4} \left (3 b^2 c^2+12 a b c d+a^2 d^2\right )+5 b d (3 b c+a d) x\right )}{x^2} \, dx}{6 c}\\ &=-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {15}{8} (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )+\frac {5}{2} b d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 c^2}\\ &=\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {15}{4} b c (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )+\frac {15}{2} b^2 c d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 b c^2}\\ &=\frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {\int \frac {\frac {15}{4} b^2 c^2 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )+\frac {15}{2} b^3 c^2 d (3 b c+a d) (b c+3 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 b^2 c^2}\\ &=\frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {1}{8} (5 b d (3 b c+a d) (b c+3 a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{16} \left (5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=\frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}+\frac {1}{4} (5 d (3 b c+a d) (b c+3 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{8} \left (5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}-\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} \sqrt {c}}+\frac {1}{4} (5 d (3 b c+a d) (b c+3 a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {5 d \left (5 b^2 c^2+10 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c}+\frac {5 d \left (9 b^2 c^2+14 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{24 c^2}-\frac {5 \left (3 b^2 c^2+12 a b c d+a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 c^2 x}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{3 x^3}-\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} \sqrt {c}}+\frac {5}{4} \sqrt {b} \sqrt {d} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 247, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (2 a b x \left (13 c^2+61 c d x-27 d^2 x^2\right )-3 b^2 x^2 \left (-11 c^2+18 c d x+4 d^2 x^2\right )+a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{24 x^3}-\frac {5 \left (b^3 c^3+15 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 \sqrt {a} \sqrt {c}}+\frac {5}{4} \sqrt {b} \sqrt {d} \left (3 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \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. \(728\) vs.
\(2(281)=562\).
time = 0.07, size = 729, normalized size = 2.15
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{3} x^{3} \sqrt {a c}+300 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c \,d^{2} x^{3} \sqrt {a c}+90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d \,x^{3} \sqrt {a c}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3} \sqrt {b d}-225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3} \sqrt {b d}-225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3} \sqrt {b d}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3} \sqrt {b d}+24 b^{2} d^{2} x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,d^{2} x^{3}+108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c d \,x^{3}-66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-244 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}-66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -16 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2}\right )}{48 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {b d}\, \sqrt {a c}}\) | \(729\) |
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 = 5.25, size = 1473, normalized size = 4.35 \begin {gather*} \left [\frac {30 \, {\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} \sqrt {b d} x^{3} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (12 \, a b^{2} c d^{2} x^{4} - 8 \, a^{3} c^{3} + 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x^{3} - {\left (33 \, a b^{2} c^{3} + 122 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 26 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a c x^{3}}, -\frac {60 \, {\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} \sqrt {-b d} x^{3} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (12 \, a b^{2} c d^{2} x^{4} - 8 \, a^{3} c^{3} + 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x^{3} - {\left (33 \, a b^{2} c^{3} + 122 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 26 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a c x^{3}}, \frac {15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 15 \, {\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} \sqrt {b d} x^{3} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 2 \, {\left (12 \, a b^{2} c d^{2} x^{4} - 8 \, a^{3} c^{3} + 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x^{3} - {\left (33 \, a b^{2} c^{3} + 122 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 26 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a c x^{3}}, \frac {15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 30 \, {\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} \sqrt {-b d} x^{3} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (12 \, a b^{2} c d^{2} x^{4} - 8 \, a^{3} c^{3} + 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x^{3} - {\left (33 \, a b^{2} c^{3} + 122 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 26 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a c x^{3}}\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 {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{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. 2357 vs.
\(2 (281) = 562\).
time = 14.70, size = 2357, normalized size = 6.95 \begin {gather*} \text {Too large to display} \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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