Optimal. Leaf size=180 \[ \frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {106, 157, 12,
95, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}+\frac {2 d \sqrt {a+b x} (3 b c-a d) (3 a d+b c)}{3 a c^2 \sqrt {c+d x} (b c-a d)^3}+\frac {2 b}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {2 d \sqrt {a+b x} (a d+3 b c)}{3 a c (c+d x)^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \int \frac {\frac {1}{2} (b c-a d)+2 b d x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a (b c-a d)}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}-\frac {4 \int \frac {-\frac {3}{4} (b c-a d)^2-\frac {1}{2} b d (3 b c+a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c (b c-a d)^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}+\frac {8 \int \frac {3 (b c-a d)^3}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^2 (b c-a d)^3}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}+\frac {\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}+\frac {2 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a c^2}\\ &=\frac {2 b}{a (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 d (3 b c+a d) \sqrt {a+b x}}{3 a c (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 d (3 b c-a d) (b c+3 a d) \sqrt {a+b x}}{3 a c^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 150, normalized size = 0.83 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (a c d^3-\frac {9 a b c d^2 (c+d x)}{a+b x}+\frac {3 a^2 d^3 (c+d x)}{a+b x}-\frac {3 b^3 c^2 (c+d x)^2}{(a+b x)^2}\right )}{3 a c^2 (-b c+a d)^3 (c+d x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{3/2} c^{5/2}} \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. \(1252\) vs.
\(2(152)=304\).
time = 0.06, size = 1253, normalized size = 6.96
method | result | size |
default | \(-\frac {-6 a^{2} b \,d^{4} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-15 \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^{3} b \,c^{2} d^{3} x +9 \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^{2} b^{2} c^{3} d^{2} x +6 b^{3} c^{2} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+12 b^{3} c^{3} d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+18 a^{2} b \,c^{2} d^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-9 \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^{2} b^{2} c \,d^{4} x^{3}+9 \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 \,b^{3} c^{2} d^{3} x^{3}-3 \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^{3} b c \,d^{4} x^{2}-9 \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^{2} b^{2} c^{2} d^{3} x^{2}+15 \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 \,b^{3} c^{3} d^{2} x^{2}+3 \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 \,b^{3} c^{4} d x +6 b^{3} c^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+3 \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^{4} d^{5} x^{2}-3 \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^{4} c^{5} x +3 \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^{4} c^{2} d^{3}-3 \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 \,b^{3} c^{5}+8 a^{2} b c \,d^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+16 a \,b^{2} c \,d^{3} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+18 a \,b^{2} c^{2} d^{2} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-6 a^{3} d^{4} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-8 a^{3} c \,d^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+3 \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^{3} b \,d^{5} x^{3}-3 \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^{4} c^{3} d^{2} x^{3}-6 \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^{4} c^{4} d \,x^{2}+6 \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^{4} c \,d^{4} x -9 \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^{3} b \,c^{3} d^{2}+9 \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^{2} b^{2} c^{4} d}{3 \sqrt {a c}\, \left (a d -b c \right )^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, c^{2} a}\) | \(1253\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 658 vs.
\(2 (152) = 304\).
time = 1.63, size = 1336, normalized size = 7.42 \begin {gather*} \left [\frac {3 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\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 b^{3} c^{5} + 9 \, a^{3} b c^{3} d^{2} - 4 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} + 8 \, a^{2} b^{2} c^{2} d^{3} - 3 \, a^{3} b c d^{4}\right )} x^{2} + {\left (6 \, a b^{3} c^{4} d + 9 \, a^{2} b^{2} c^{3} d^{2} + 4 \, a^{3} b c^{2} d^{3} - 3 \, a^{4} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3} + {\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{3} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{2} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x\right )}}, \frac {3 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\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 b^{3} c^{5} + 9 \, a^{3} b c^{3} d^{2} - 4 \, a^{4} c^{2} d^{3} + {\left (3 \, a b^{3} c^{3} d^{2} + 8 \, a^{2} b^{2} c^{2} d^{3} - 3 \, a^{3} b c d^{4}\right )} x^{2} + {\left (6 \, a b^{3} c^{4} d + 9 \, a^{2} b^{2} c^{3} d^{2} + 4 \, a^{3} b c^{2} d^{3} - 3 \, a^{4} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3} + {\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{3} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{2} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x\right )}}\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 {1}{x \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, 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. 520 vs.
\(2 (152) = 304\).
time = 2.09, size = 520, normalized size = 2.89 \begin {gather*} \frac {4 \, \sqrt {b d} b^{4}}{{\left (a b^{2} c^{2} {\left | b \right |} - 2 \, a^{2} b c d {\left | b \right |} + a^{3} d^{2} {\left | b \right |}\right )} {\left (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 )}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (8 \, b^{6} c^{5} d^{4} {\left | b \right |} - 19 \, a b^{5} c^{4} d^{5} {\left | b \right |} + 14 \, a^{2} b^{4} c^{3} d^{6} {\left | b \right |} - 3 \, a^{3} b^{3} c^{2} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{9} d - 5 \, a b^{6} c^{8} d^{2} + 10 \, a^{2} b^{5} c^{7} d^{3} - 10 \, a^{3} b^{4} c^{6} d^{4} + 5 \, a^{4} b^{3} c^{5} d^{5} - a^{5} b^{2} c^{4} d^{6}} + \frac {3 \, {\left (3 \, b^{7} c^{6} d^{3} {\left | b \right |} - 10 \, a b^{6} c^{5} d^{4} {\left | b \right |} + 12 \, a^{2} b^{5} c^{4} d^{5} {\left | b \right |} - 6 \, a^{3} b^{4} c^{3} d^{6} {\left | b \right |} + a^{4} b^{3} c^{2} d^{7} {\left | b \right |}\right )}}{b^{7} c^{9} d - 5 \, a b^{6} c^{8} d^{2} + 10 \, a^{2} b^{5} c^{7} d^{3} - 10 \, a^{3} b^{4} c^{6} d^{4} + 5 \, a^{4} b^{3} c^{5} d^{5} - a^{5} b^{2} c^{4} d^{6}}\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} a c^{2} {\left | b \right |}} \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 {1}{x\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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