Optimal. Leaf size=77 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {2 d \sqrt {a+b x}}{c \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 93, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {2 d \sqrt {a+b x}}{c \sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx &=-\frac {2 d \sqrt {a+b x}}{c (b c-a d) \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c}\\ &=-\frac {2 d \sqrt {a+b x}}{c (b c-a d) \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}\\ &=-\frac {2 d \sqrt {a+b x}}{c (b c-a d) \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 1.00 \[ \frac {2 d \sqrt {a+b x}}{c \sqrt {c+d x} (a d-b c)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 346, normalized size = 4.49 \[ \left [-\frac {4 \, \sqrt {b x + a} \sqrt {d x + c} a c d - {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right )}{2 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x\right )}}, -\frac {2 \, \sqrt {b x + a} \sqrt {d x + c} a c d - {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right )}{a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 140, normalized size = 1.82 \[ -\frac {2 \, \sqrt {b x + a} b^{2} d}{{\left (b c^{2} {\left | b \right |} - a c d {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 243, normalized size = 3.16 \[ \frac {\left (-a \,d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+b c d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-a c d \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+b \,c^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, d \right ) \sqrt {b x +a}}{\left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, c} \]
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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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