Optimal. Leaf size=191 \[ \frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}+\frac {3}{4} \sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)-3 \sqrt {a} \sqrt {c} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \[ \frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}+\frac {3}{4} \sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)-3 \sqrt {a} \sqrt {c} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 97
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{2} (b c+a d)+3 b d x\right )}{x} \, dx\\ &=\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {\sqrt {c+d x} \left (3 a d (b c+a d)+\frac {3}{2} b d (b c+3 a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 d}\\ &=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {\int \frac {3 a b c d (b c+a d)+\frac {3}{4} b d \left (b^2 c^2+6 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d}\\ &=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac {1}{2} (3 a c (b c+a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{8} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}+(3 a c (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 )+\frac {\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b}\\ &=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b}\\ &=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 197, normalized size = 1.03 \[ \frac {3 \sqrt {c+d x} \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (a (5 d x-4 c)+b x (5 c+2 d x))}{4 x}-3 \sqrt {a} \sqrt {c} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 3.61, size = 1073, normalized size = 5.62 \[ \left [\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \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 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, -\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 6 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \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 ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}, \frac {24 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \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 ) + 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, \frac {12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \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 ) - 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.06, size = 593, normalized size = 3.10 \[ \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b} + \frac {5 \, b c d^{2} {\left | b \right |} + 3 \, a d^{3} {\left | b \right |}}{b d^{2}}\right )} - \frac {24 \, {\left (\sqrt {b d} a b^{2} c^{2} {\left | b \right |} + \sqrt {b d} a^{2} b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} a b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{3} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b c d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}} - \frac {3 \, {\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} + 6 \, \sqrt {b d} a b c d {\left | b \right |} + \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b d}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 489, normalized size = 2.56 \[ \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-12 \sqrt {b d}\, a^{2} c d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+3 \sqrt {a c}\, a^{2} d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-12 \sqrt {b d}\, a b \,c^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+18 \sqrt {a c}\, a b c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 \sqrt {a c}\, b^{2} c^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b d \,x^{2}+10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a d x +10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b c x -8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a c \right )}{8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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