Optimal. Leaf size=66 \[ \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {47, 63, 217, 206} \[ \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^{3/2}} \, dx &=-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}+\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b}\\ &=-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}+\frac {(2 d) \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^2}\\ &=-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}+\frac {(2 d) \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^2}\\ &=-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 99, normalized size = 1.50 \[ \frac {2 \left (\sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )-\frac {b (c+d x)}{\sqrt {a+b x}}\right )}{b^2 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 241, normalized size = 3.65 \[ \left [\frac {{\left (b x + a\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} x + a b\right )}}, -\frac {{\left (b x + a\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c}}{b^{2} x + a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 131, normalized size = 1.98 \[ -\frac {{\left (\frac {\sqrt {b d} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} + \frac {4 \, {\left (\sqrt {b d} b c - \sqrt {b d} a d\right )}}{b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}\right )} {\left | b \right |}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x +c}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 = -0.02 \[ \int \frac {\sqrt {c+d\,x}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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