Optimal. Leaf size=194 \[ \frac {d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{4 a^2 c^3 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} (5 a d+3 b c)}{4 a^2 c^2 x \sqrt {c+d x}}-\frac {3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}} \]
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Rubi [A] time = 0.16, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {103, 151, 152, 12, 93, 208} \[ -\frac {3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2}}+\frac {d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{4 a^2 c^3 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} (5 a d+3 b c)}{4 a^2 c^2 x \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx &=-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}-\frac {\int \frac {\frac {1}{2} (3 b c+5 a d)+2 b d x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 a c}\\ &=-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}+\frac {\int \frac {\frac {3}{4} \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )+\frac {1}{2} b d (3 b c+5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 a^2 c^2}\\ &=\frac {d (3 b c-5 a d) (b c+3 a d) \sqrt {a+b x}}{4 a^2 c^3 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}-\frac {\int -\frac {3 (b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 c^3 (b c-a d)}\\ &=\frac {d (3 b c-5 a d) (b c+3 a d) \sqrt {a+b x}}{4 a^2 c^3 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}+\frac {\left (3 \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^2 c^3}\\ &=\frac {d (3 b c-5 a d) (b c+3 a d) \sqrt {a+b x}}{4 a^2 c^3 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}+\frac {\left (3 \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^2 c^3}\\ &=\frac {d (3 b c-5 a d) (b c+3 a d) \sqrt {a+b x}}{4 a^2 c^3 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}-\frac {3 \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 186, normalized size = 0.96 \[ \frac {\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \left (a^2 d \left (2 c^2-5 c d x-15 d^2 x^2\right )+2 a b c \left (-c^2+c d x+2 d^2 x^2\right )+3 b^2 c^2 x (c+d x)\right )}{x^2 \sqrt {c+d x}}-3 \left (-5 a^3 d^3+3 a^2 b c d^2+a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.52, size = 664, normalized size = 3.42 \[ \left [\frac {3 \, {\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} + {\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} b c^{4} - 2 \, a^{3} c^{3} d - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{2} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left ({\left (a^{3} b c^{5} d - a^{4} c^{4} d^{2}\right )} x^{3} + {\left (a^{3} b c^{6} - a^{4} c^{5} d\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} + {\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} b c^{4} - 2 \, a^{3} c^{3} d - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{2} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left ({\left (a^{3} b c^{5} d - a^{4} c^{4} d^{2}\right )} x^{3} + {\left (a^{3} b c^{6} - a^{4} c^{5} d\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.90, size = 1105, normalized size = 5.70 \[ -\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b c^{4} {\left | b \right |} - a c^{3} d {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {3 \, {\left (\sqrt {b d} b^{4} c^{2} + 2 \, \sqrt {b d} a b^{3} c d + 5 \, \sqrt {b d} a^{2} b^{2} d^{2}\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 )}{4 \, \sqrt {-a b c d} a^{2} b c^{3} {\left | b \right |}} + \frac {3 \, \sqrt {b d} b^{10} c^{5} - 5 \, \sqrt {b d} a b^{9} c^{4} d - 10 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} + 30 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} - 25 \, \sqrt {b d} a^{4} b^{6} c d^{4} + 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 9 \, \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^{8} c^{4} - 16 \, \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 b^{7} c^{3} d + 38 \, \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^{6} c^{2} d^{2} + 8 \, \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^{3} b^{5} c d^{3} - 21 \, \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^{4} b^{4} d^{4} + 9 \, \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^{6} c^{3} + 27 \, \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 b^{5} c^{2} d + 23 \, \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^{4} c d^{2} + 21 \, \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^{3} b^{3} d^{3} - 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 )}^{6} b^{4} c^{2} - 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 )}^{6} a b^{3} c d - 7 \, \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 )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{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} b^{2} c - 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 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}\right )}^{2} a^{2} c^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 683, normalized size = 3.52 \[ -\frac {\sqrt {b x +a}\, \left (15 a^{3} 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 )-9 a^{2} b 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^{2} 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^{3} 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 )+15 a^{3} 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 )-9 a^{2} b \,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 a \,b^{2} c^{3} d \,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^{3} 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 )-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{3} x^{2}+8 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{2} x^{2}+6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d \,x^{2}-10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \,d^{2} x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d x +6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} d -4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3}\right )}{8 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, a^{2} c^{3} x^{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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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