Optimal. Leaf size=77 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}-\frac {2 c \sqrt {a+b x}}{d \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {78, 63, 217, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}-\frac {2 c \sqrt {a+b x}}{d \sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx &=-\frac {2 c \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}+\frac {\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 c \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b d}\\ &=-\frac {2 c \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b d}\\ &=-\frac {2 c \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 110, normalized size = 1.43 \[ \frac {2 \left (b c \sqrt {d} \sqrt {a+b x}-(b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{b d^{3/2} \sqrt {c+d x} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 335, normalized size = 4.35 \[ \left [-\frac {4 \, \sqrt {b x + a} \sqrt {d x + c} b c d - {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{2 \, {\left (b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x\right )}}, -\frac {2 \, \sqrt {b x + a} \sqrt {d x + c} b c d + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 108, normalized size = 1.40 \[ -\frac {2 \, {\left (\frac {\sqrt {b x + a} b^{3} c {\left | b \right |}}{{\left (b^{3} c d - a b^{2} d^{2}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left | b \right |} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 251, normalized size = 3.26 \[ \frac {\sqrt {b x +a}\, \left (a \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-b c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+a c d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-b \,c^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, c \right )}{\sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, d} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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