Optimal. Leaf size=185 \[ -\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {3 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 185, 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 = {105, 157, 12,
95, 214} \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 )}{a^{5/2} c^{5/2}}-\frac {d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {\int \frac {\frac {3}{2} (b c+a d)+2 b d x}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{a c}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \int \frac {\frac {3}{4} (b c-a d) (b c+a d)+\frac {1}{2} b d (3 b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {4 \int -\frac {3 (b c-a d)^2 (b c+a d)}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 c^2 (b c-a d)^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2 c^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (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 )}{a^2 c^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {3 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 165, normalized size = 0.89 \begin {gather*} \frac {-3 b^3 c^2 x (c+d x)-a^3 d^2 (c+3 d x)+a^2 b d \left (2 c^2+c d x-3 d^2 x^2\right )+a b^2 c \left (-c^2+c d x+2 d^2 x^2\right )}{a^2 c^2 (b c-a d)^2 x \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} 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. \(896\) vs.
\(2(161)=322\).
time = 0.07, size = 897, normalized size = 4.85
method | result | size |
default | \(\frac {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^{4} 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^{2} b^{2} c \,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 \,b^{3} c^{2} d^{2} 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 \,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^{4} d^{4} 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^{2} b^{2} c^{2} 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 ) b^{4} c^{4} 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^{4} c \,d^{3} 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^{3} b \,c^{2} d^{2} 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^{2} b^{2} c^{3} d 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 \,b^{3} c^{4} x -6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{2}-6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{2}-6 a^{3} d^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d x -6 b^{3} c^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-2 a^{3} c \,d^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+4 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d -2 a \,b^{2} c^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}}{2 c^{2} a^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \left (a d -b c \right )^{2} x \sqrt {a c}\, \sqrt {b x +a}\, \sqrt {d x +c}}\) | \(897\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs.
\(2 (161) = 322\).
time = 2.06, size = 940, normalized size = 5.08 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} 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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x\right )}}, -\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} 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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\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^{2} \left (a + b x\right )^{\frac {3}{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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 817 vs.
\(2 (161) = 322\).
time = 3.30, size = 817, normalized size = 4.42 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b^{2} c^{4} {\left | b \right |} - 2 \, a b c^{3} d {\left | b \right |} + a^{2} c^{2} d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{8} c^{4} - 8 \, \sqrt {b d} a b^{7} c^{3} d + 8 \, \sqrt {b d} a^{2} b^{6} c^{2} d^{2} - 4 \, \sqrt {b d} a^{3} b^{5} c d^{3} + \sqrt {b d} a^{4} b^{4} d^{4} - 6 \, \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^{6} c^{3} - 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} a^{2} b^{4} c d^{2} + 3 \, \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 )}^{4} b^{4} c^{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 )}^{4} a^{2} b^{2} d^{2}\right )}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{4} c^{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} a b^{3} c 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^{2} b^{2} d^{2} + 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} 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 )}^{6}\right )} {\left (a^{2} b c^{3} {\left | b \right |} - a^{3} c^{2} d {\left | b \right |}\right )}} + \frac {3 \, {\left (\sqrt {b d} b^{3} c + \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} a^{2} b 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^2\,{\left (a+b\,x\right )}^{3/2}\,{\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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