Optimal. Leaf size=213 \[ \frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {103, 159, 163,
65, 223, 212, 95, 214} \begin {gather*} -2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac {1}{8} \sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+8 a c-\frac {b c^2}{d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx &=\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {1}{3} \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a c-\frac {3}{2} (b c+a d) x\right )}{x} \, dx\\ &=\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {\int \frac {\sqrt {c+d x} \left (-6 a^2 c d+\frac {3}{4} \left (b^2 c^2-8 a b c d-a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {\int \frac {-6 a^2 b c^2 d+\frac {3}{8} (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (a^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (2 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^2 d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b^2 d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 195, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2+2 a b d (19 c+7 d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 b d}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {\left (b^3 c^3-9 a b^2 c^2 d-9 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} 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. \(502\) vs.
\(2(169)=338\).
time = 0.07, size = 503, normalized size = 2.36
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-16 b^{2} d^{2} x^{2} \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} d^{3}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b c \,d^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c^{2} d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3}+48 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,d^{2} x -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2}-76 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2}\right )}{48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b d}\) | \(503\) |
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 = 5.96, size = 1193, normalized size = 5.60 \begin {gather*} \left [\frac {48 \, \sqrt {a c} a b^{2} c d^{2} \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 ) + 3 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} \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 ) + 4 \, {\left (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 38 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 14 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{2} d^{2}}, \frac {24 \, \sqrt {a c} a b^{2} c d^{2} \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 ) + 3 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} \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 (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 38 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 14 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{2} d^{2}}, \frac {96 \, \sqrt {-a c} a b^{2} c d^{2} \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 ) + 3 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} \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 ) + 4 \, {\left (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 38 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 14 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{2} d^{2}}, \frac {48 \, \sqrt {-a c} a b^{2} c d^{2} \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 ) + 3 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} \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 (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 38 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 14 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{2} d^{2}}\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 {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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