Optimal. Leaf size=279 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}+\frac {(b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{9/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {101, 156, 12,
95, 214} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{96 a^2 c^3 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{192 a^3 c^4 x}+\frac {(b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{24 a c^2 x^3}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}+\frac {\int \frac {\frac {1}{2} (b c-7 a d)-3 b d x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{4 c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {\int \frac {\frac {1}{4} \left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right )+b d (b c-7 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}+\frac {\int \frac {\frac {1}{8} \left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right )+\frac {1}{4} b d \left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^2 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}-\frac {\int \frac {3 (b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^3 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}-\frac {\left ((b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^3 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}-\frac {\left ((b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a^3 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}+\frac {(b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 10.21, size = 244, normalized size = 0.87 \begin {gather*} \frac {-48 (a+b x)^{3/2} \sqrt {c+d x}+\frac {8 (5 b c+7 a d) x (a+b x)^{3/2} \sqrt {c+d x}}{a c}-\frac {2 \left (15 b^2 c^2+22 a b c d+35 a^2 d^2\right ) x^2 (a+b x)^{3/2} \sqrt {c+d x}}{a^2 c^2}+\frac {3 \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) x^3 \left (\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x}+(b c-a d) x \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{a^{5/2} c^{7/2}}}{192 a c x^4} \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. \(592\) vs.
\(2(241)=482\).
time = 0.07, size = 593, normalized size = 2.13
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3} x^{3}+50 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2} x^{3}+34 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3} x^{3}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c \,d^{2} x^{2}-24 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{2} d \,x^{2}-20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{2} d x +16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{3} x +96 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{3} c^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
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 = 3.92, size = 570, normalized size = 2.04 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{5} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{5} x^{4}}\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}}{x^{5} \sqrt {c + d x}}\, 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. 3690 vs.
\(2 (241) = 482\).
time = 3.16, size = 3690, normalized size = 13.23 \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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