Optimal. Leaf size=102 \[ -\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.05, antiderivative size = 102, 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} \[ \frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)} \]
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 (c+d x)^{5/2}} \, dx &=-\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {\int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{c}\\ &=-\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}+\frac {a \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c^2}\\ &=-\frac {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}+\frac {(2 a) \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 {2 d (a+b x)^{3/2}}{3 c (b c-a d) (c+d x)^{3/2}}+\frac {2 \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 102, normalized size = 1.00 \[ \frac {2 \sqrt {a+b x} (b c (3 c+2 d x)-a d (4 c+3 d x))}{3 c^2 (c+d x)^{3/2} (b c-a d)}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.58, size = 479, normalized size = 4.70 \[ \left [\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}, \frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.61, size = 243, normalized size = 2.38 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (2 \, b^{4} c^{3} d^{2} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}} + \frac {3 \, {\left (b^{5} c^{4} d {\left | b \right |} - 2 \, a b^{4} c^{3} d^{2} {\left | b \right |} + a^{2} b^{3} c^{2} d^{3} {\left | b \right |}\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, \sqrt {b d} a b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 430, normalized size = 4.22 \[ -\frac {\left (3 a^{2} d^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a b c \,d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+6 a^{2} c \,d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 a b \,c^{2} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{2} c^{2} d \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a b \,c^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{2} x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d x -8 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c d +6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2}\right ) \sqrt {b x +a}}{3 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} c^{2}} \]
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 \[ \int \frac {\sqrt {a+b\,x}}{x\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x}}{x \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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