Optimal. Leaf size=167 \[ \frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )+d x (b c-3 a d) (b c-a d)\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^2}-\frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac {2 a x^2}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.13, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {98, 143, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )+d x (b c-3 a d) (b c-a d)\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^2}-\frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac {2 a x^2}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 143
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \int \frac {x \left (2 a c+\frac {1}{2} (-b c+3 a d) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{b (b c-a d)}\\ &=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2 d^2}\\ &=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3 d^2}\\ &=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3 d^2}\\ &=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {3 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 195, normalized size = 1.17 \begin {gather*} \frac {\frac {\sqrt {d} \left (3 a^3 d^2 (c+d x)+a^2 b d \left (-2 c^2-c d x+d^2 x^2\right )+a b^2 c \left (3 c^2-c d x-2 d^2 x^2\right )+b^3 c^2 x (3 c+d x)\right )}{\sqrt {a+b x} (b c-a d)^2}-\frac {3 \sqrt {b c-a d} (a d+b c) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b}}{b^2 d^{5/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 222, normalized size = 1.33 \begin {gather*} \frac {\sqrt {a+b x} \left (-\frac {3 a^3 d^3 (c+d x)}{a+b x}+\frac {2 a^3 b d^2 (c+d x)^2}{(a+b x)^2}+\frac {3 a^2 b c d^2 (c+d x)}{a+b x}+\frac {3 b^3 c^3 (c+d x)}{a+b x}-\frac {3 a b^2 c^2 d (c+d x)}{a+b x}-2 b^2 c^3 d\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^2 \left (\frac {b (c+d x)}{a+b x}-d\right )}-\frac {3 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 910, normalized size = 5.45 \begin {gather*} \left [\frac {3 \, {\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} + {\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^{2} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x\right )} \sqrt {b d} \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 (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x\right )}}, \frac {3 \, {\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} + {\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^{2} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x\right )} \sqrt {-b d} \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 (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.81, size = 342, normalized size = 2.05 \begin {gather*} \frac {4 \, \sqrt {b d} a^{3}}{{\left (b^{2} c {\left | b \right |} - a b d {\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 {\sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{2} d^{2} - 2 \, a b^{5} c d^{3} + a^{2} b^{4} d^{4}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\left | b \right |}} + \frac {3 \, b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 3 \, a^{2} b^{5} c d^{3} - a^{3} b^{4} d^{4}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\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 c + \sqrt {b d} a d\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 )}{2 \, b^{2} d^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 906, normalized size = 5.43 \begin {gather*} -\frac {3 a^{3} b \,d^{4} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a^{2} b^{2} c \,d^{3} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a \,b^{3} c^{2} d^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 b^{4} c^{3} d \,x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{4} d^{4} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-6 a^{2} b^{2} c^{2} d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 b^{4} c^{4} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{4} c \,d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a^{3} b \,c^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a^{2} b^{2} c^{3} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a \,b^{3} c^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x^{2}-2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d \,x^{2}-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3} x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2} x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{3} x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} c \,d^{2}+4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,c^{2} d -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, 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 {b d}\, \sqrt {b x +a}\, \sqrt {d x +c}\, b^{2} d^{2}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\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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