Optimal. Leaf size=345 \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}-\frac {(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{40 a c x^4} \]
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Rubi [A] time = 0.32, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {97, 151, 12, 93, 208} \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac {(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{40 a c x^4}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 97
Rule 151
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}+\frac {1}{5} \int \frac {\frac {1}{2} (b c+a d)+b d x}{x^5 \sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}-\frac {\int \frac {\frac {1}{4} \left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right )+\frac {3}{2} b d (b c+a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}+\frac {\int \frac {\frac {1}{8} (b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right )+\frac {1}{2} b d \left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^2 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}-\frac {\int \frac {\frac {1}{16} \left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right )+\frac {1}{8} b d (b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\int \frac {15 (b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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 )}{128 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}-\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 253, normalized size = 0.73 \[ -\frac {\frac {8 x^2 (a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2+38 a b c d+35 b^2 c^2\right )}{a^2 c^2}+\frac {15 x^3 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+a d x+b c x)\right )}{a^{7/2} c^{7/2}}-\frac {336 x (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{a c}+384 (a+b x)^{3/2} (c+d x)^{3/2}}{1920 a c x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 11.38, size = 730, normalized size = 2.12 \[ \left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{5} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{5} x^{5}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 967, normalized size = 2.80 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 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 )-75 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 )-75 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 )+105 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 )-210 \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}+68 \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}-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-44 \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}-44 \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}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}+32 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x +96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4} x^{5}} \]
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(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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