Optimal. Leaf size=113 \[ -\frac {(b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}+\frac {\sqrt {a+b x} (b c-3 a d)}{a c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}} \]
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Rubi [A] time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} (b c-3 a d)}{a c^2 \sqrt {c+d x}}-\frac {(b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{3/2}} \, dx &=-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}}-\frac {\left (-\frac {b c}{2}+\frac {3 a d}{2}\right ) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{a c}\\ &=\frac {(b c-3 a d) \sqrt {a+b x}}{a c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}}+\frac {(b c-3 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^2}\\ &=\frac {(b c-3 a d) \sqrt {a+b x}}{a c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}}+\frac {(b c-3 a d) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^2}\\ &=\frac {(b c-3 a d) \sqrt {a+b x}}{a c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{a c x \sqrt {c+d x}}-\frac {(b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.73 \begin {gather*} \frac {(3 a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {\sqrt {a+b x} (c+3 d x)}{c^2 x \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 118, normalized size = 1.04 \begin {gather*} \frac {(3 a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {\sqrt {a+b x} \left (\frac {2 c d (a+b x)}{c+d x}-3 a d+b c\right )}{c^2 \sqrt {c+d x} \left (\frac {c (a+b x)}{c+d x}-a\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.93, size = 330, normalized size = 2.92 \begin {gather*} \left [-\frac {{\left ({\left (b c d - 3 \, a d^{2}\right )} x^{2} + {\left (b c^{2} - 3 \, a c d\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 ) + 4 \, {\left (3 \, a c d x + a c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (a c^{3} d x^{2} + a c^{4} x\right )}}, \frac {{\left ({\left (b c d - 3 \, a d^{2}\right )} x^{2} + {\left (b c^{2} - 3 \, a c d\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 ) - 2 \, {\left (3 \, a c d x + a c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a c^{3} d x^{2} + a c^{4} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.21, size = 457, normalized size = 4.04 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2} d}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{2} {\left | b \right |}} - \frac {{\left (\sqrt {b d} b^{3} c - 3 \, \sqrt {b d} a b^{2} 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^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} 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^{3} 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^{2} 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^{2} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 267, normalized size = 2.36 \begin {gather*} \frac {\sqrt {b x +a}\, \left (3 a \,d^{2} x^{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 )-b c d \,x^{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 )+3 a 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 )-b \,c^{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 )-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, c \right )}{2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, c^{2} x} \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}}{x^2\,{\left (c+d\,x\right )}^{3/2}} \,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}}{x^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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