Optimal. Leaf size=185 \[ \frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac {d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}} \]
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Rubi [A] time = 0.15, 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 = {103, 152, 12, 93, 208} \[ -\frac {d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}+\frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 208
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)) \operatorname {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 = 0.21, size = 165, normalized size = 0.89 \[ \frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac {a^3 \left (-d^2\right ) (c+3 d x)+a^2 b d \left (2 c^2+c d x-3 d^2 x^2\right )+a b^2 c \left (-c^2+c d x+2 d^2 x^2\right )-3 b^3 c^2 x (c+d x)}{a^2 c^2 x \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.19, size = 940, normalized size = 5.08 \[ \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 ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 14.86, size = 817, normalized size = 4.42 \[ -\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 |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 897, normalized size = 4.85 \[ \frac {3 a^{3} b \,d^{4} x^{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 )-3 a^{2} b^{2} c \,d^{3} x^{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 )-3 a \,b^{3} c^{2} d^{2} x^{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 )+3 b^{4} c^{3} d \,x^{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 )+3 a^{4} 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^{2} 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 )+3 b^{4} c^{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 )+3 a^{4} 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 )-3 a^{3} 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 )-3 a^{2} 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 \,b^{3} c^{4} 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 )}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c \,d^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{2} d \,x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c \,d^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,c^{2} d -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{3}}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (a d -b c \right )^{2} \sqrt {a c}\, \sqrt {b x +a}\, \sqrt {d x +c}\, a^{2} c^{2} x} \]
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 \[ \int \frac {1}{x^2\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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