Optimal. Leaf size=124 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{3 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {104, 152, 12, 93, 208} \[ -\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{3 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 104
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx &=-\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (b c-a d)+b d x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 c (b c-a d)}\\ &=-\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {4 \int \frac {3 (b c-a d)^2}{4 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 c^2 (b c-a d)^2}\\ &=-\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c^2}\\ &=-\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \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 \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \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 = 0.18, size = 125, normalized size = 1.01 \[ \frac {2 \left (\frac {3 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}+\frac {d \sqrt {a+b x} (b c (6 c+5 d x)-a d (4 c+3 d x))}{c (c+d x)^{3/2} (b c-a d)}\right )}{3 c (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.31, size = 678, normalized size = 5.47 \[ \left [\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 271, normalized size = 2.19 \[ -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{4} c^{3} d^{3} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}} + \frac {3 \, {\left (2 \, b^{5} c^{4} d^{2} {\left | b \right |} - 3 \, a b^{4} c^{3} d^{3} {\left | b \right |} + a^{2} b^{3} c^{2} d^{4} {\left | b \right |}\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 586, normalized size = 4.73 \[ -\frac {\sqrt {b x +a}\, \left (3 a^{2} d^{4} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 a b c \,d^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{2} c^{2} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+6 a^{2} c \,d^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-12 a b \,c^{2} d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+6 b^{2} c^{3} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{2} c^{2} d^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 a b \,c^{3} d \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{2} c^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,d^{3} x +10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b c \,d^{2} x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a c \,d^{2}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b \,c^{2} d \right )}{3 \sqrt {a c}\, \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{2}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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