Optimal. Leaf size=77 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}} \]
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Rubi [A]
time = 0.02, 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, 95, 214}
\begin {gather*} -\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}+\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}+\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}-\frac {(b c-a d) \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 = 0.46, size = 121, normalized size = 1.57 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}-\frac {\sqrt {\frac {b}{d}} \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )}{\sqrt {a} \sqrt {b} c^{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. \(146\) vs.
\(2(61)=122\).
time = 0.08, size = 147, normalized size = 1.91
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b c x -2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{2 c \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x \sqrt {a c}}\) | \(147\) |
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 = 1.59, size = 252, normalized size = 3.27 \begin {gather*} \left [-\frac {\sqrt {a c} {\left (b c - a d\right )} x \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 ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} a c}{4 \, a c^{2} x}, \frac {\sqrt {-a c} {\left (b c - a d\right )} x \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 ) - 2 \, \sqrt {b x + a} \sqrt {d x + c} a c}{2 \, a c^{2} x}\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 {\sqrt {a + b x}}{x^{2} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs.
\(2 (61) = 122\).
time = 1.45, size = 410, normalized size = 5.32 \begin {gather*} -\frac {b {\left (\frac {{\left (\sqrt {b d} b^{2} c - \sqrt {b d} a b d\right )} \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} b c} + \frac {2 \, {\left (\sqrt {b d} b^{4} c^{2} - 2 \, \sqrt {b d} a b^{3} c d + \sqrt {b d} a^{2} b^{2} d^{2} - \sqrt {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} b^{2} c - \sqrt {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} a b d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}\right )} c}\right )}}{{\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.18, size = 439, normalized size = 5.70 \begin {gather*} \frac {\frac {\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{4}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a}\,c^{3/2}\,d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {b^2}{4\,c\,d}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{4}-\frac {3\,a\,b\,c\,d}{4}+\frac {b^2\,c^2}{4}\right )}{a\,c^2\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {\left (a\,d+b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}-\frac {d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{4\,c\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2}-\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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