Optimal. Leaf size=345 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}-\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}} \]
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Rubi [A]
time = 0.22, 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 = {99, 156, 12, 95,
214} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 99
Rule 156
Rule 214
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 ) \text {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 = 0.84, size = 271, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 b^4 c^4 x^4+10 a b^3 c^3 x^3 (7 c+4 d x)-2 a^2 b^2 c^2 x^2 \left (28 c^2+11 c d x-17 d^2 x^2\right )+2 a^3 b c x \left (24 c^3+8 c^2 d x-11 c d^2 x^2+20 d^3 x^3\right )+a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )\right )}{1920 a^4 c^4 x^5}-\frac {(b c-a d)^2 \left (7 b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+7 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{9/2} c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs.
\(2(301)=602\).
time = 0.07, size = 813, normalized size = 2.36
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5}-75 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5}-75 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+80 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}+68 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}-44 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}-44 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}+32 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x +96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x +768 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{4} c^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{5} \sqrt {a c}}\) | \(813\) |
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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Fricas [A]
time = 4.96, size = 730, normalized size = 2.12 \begin {gather*} \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 ] \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 {\sqrt {a + b x} \sqrt {c + d x}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5928 vs.
\(2 (301) = 602\).
time = 38.65, size = 5928, normalized size = 17.18 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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