Optimal. Leaf size=273 \[ \frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]
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Rubi [A] time = 0.15, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {(b c+a d) \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 a c}\\ &=\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 \left (b^2-\frac {a^2 d^2}{c^2}\right )\right ) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 a}\\ &=\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{32 a c^3}\\ &=\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac {\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a^2 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^3 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^3 c^3}\\ &=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {\left (b^2-\frac {a^2 d^2}{c^2}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac {3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 228, normalized size = 0.84 \begin {gather*} \frac {(a d+b c) \left (16 a^{5/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{5/2}+x (b c-a d) \left (8 a^{5/2} \sqrt {c} \sqrt {a+b x} (c+d x)^{5/2}+x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )\right )\right )}{128 a^{7/2} c^{7/2} x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 305, normalized size = 1.12 \begin {gather*} \frac {3 (a d-b c)^4 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{7/2}}-\frac {\sqrt {c+d x} (a d-b c)^4 \left (\frac {15 a^5 d (c+d x)^4}{(a+b x)^4}+\frac {15 a^4 b c (c+d x)^4}{(a+b x)^4}-\frac {70 a^4 c d (c+d x)^3}{(a+b x)^3}-\frac {70 a^3 b c^2 (c+d x)^3}{(a+b x)^3}+\frac {128 a^3 c^2 d (c+d x)^2}{(a+b x)^2}-\frac {128 a^2 b c^3 (c+d x)^2}{(a+b x)^2}+\frac {70 a^2 c^3 d (c+d x)}{a+b x}+\frac {70 a b c^4 (c+d x)}{a+b x}-15 a c^4 d-15 b c^5\right )}{640 a^3 c^3 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 18.46, size = 720, normalized size = 2.64 \begin {gather*} \left [\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{4} c^{4} x^{5}}, -\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{4} c^{4} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 967, normalized size = 3.54 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 a^{5} d^{5} x^{5} \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 )-45 a^{4} b c \,d^{4} x^{5} \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 )+30 a^{3} b^{2} c^{2} d^{3} x^{5} \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 )+30 a^{2} b^{3} c^{3} d^{2} x^{5} \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 )-45 a \,b^{4} c^{4} d \,x^{5} \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 )+15 b^{5} c^{5} x^{5} \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 )-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-36 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-52 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}-52 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}-544 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}-352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x -352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x -256 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{1280 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{3} x^{5}} \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,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 {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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