Optimal. Leaf size=165 \[ \frac {\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} \sqrt {d}}-2 \sqrt {a} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}+\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 b} \]
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Rubi [A] time = 0.14, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} \sqrt {d}}-2 \sqrt {a} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}+\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 101
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x} \, dx &=\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}-\frac {1}{2} \int \frac {\sqrt {c+d x} \left (-2 a c+\frac {1}{2} (-3 b c-a d) x\right )}{x \sqrt {a+b x}} \, dx\\ &=\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}-\frac {\int \frac {-2 a b c^2+\frac {1}{4} \left (-3 b^2 c^2-6 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b}\\ &=\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}+\left (a c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b}\\ &=\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}+\left (2 a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^2}\\ &=\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}-2 \sqrt {a} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^2}\\ &=\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b}+\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2}-2 \sqrt {a} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 b^2 c^2+6 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 )}{4 b^{3/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 195, normalized size = 1.18 \begin {gather*} \frac {\sqrt {b c-a d} \left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )+b \sqrt {d} \left (\sqrt {a+b x} (c+d x) (a d+5 b c+2 b d x)-8 \sqrt {a} b c^{3/2} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{4 b^2 \sqrt {d} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 241, normalized size = 1.46 \begin {gather*} \frac {\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} \sqrt {d}}-\frac {\sqrt {a+b x} \left (-\frac {a^2 d^3 (a+b x)}{c+d x}-a^2 b d^2+\frac {3 b^2 c^2 d (a+b x)}{c+d x}+6 a b^2 c d-\frac {2 a b c d^2 (a+b x)}{c+d x}-5 b^3 c^2\right )}{4 b \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-2 \sqrt {a} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 6.36, size = 985, normalized size = 5.97 \begin {gather*} \left [\frac {8 \, \sqrt {a c} b^{2} c d \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 ) - {\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + 5 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d}, \frac {4 \, \sqrt {a c} b^{2} c d \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 ) - {\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + 5 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d}, \frac {16 \, \sqrt {-a c} b^{2} c d \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 ) - {\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + 5 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d}, \frac {8 \, \sqrt {-a c} b^{2} c d \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 ) - {\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + 5 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d}\right ] \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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maple [B] time = 0.02, size = 388, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (\sqrt {a c}\, a^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+8 \sqrt {b d}\, a b \,c^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-6 \sqrt {a c}\, a b c d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 \sqrt {a c}\, b^{2} c^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b d x -2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a d -10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b c \right )}{8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,x}\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x} \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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