Optimal. Leaf size=136 \[ \frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {7776 b^3 \sqrt [6]{a+b x}}{1729 (b c-a d)^4 \sqrt [6]{c+d x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} \frac {7776 b^3 \sqrt [6]{a+b x}}{1729 \sqrt [6]{c+d x} (b c-a d)^4}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (c+d x)^{7/6} (b c-a d)^3}+\frac {108 b \sqrt [6]{a+b x}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{19 (c+d x)^{19/6} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx &=\frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {(18 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx}{19 (b c-a d)}\\ &=\frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {\left (216 b^2\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx}{247 (b c-a d)^2}\\ &=\frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {\left (1296 b^3\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{1729 (b c-a d)^3}\\ &=\frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {7776 b^3 \sqrt [6]{a+b x}}{1729 (b c-a d)^4 \sqrt [6]{c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 93, normalized size = 0.68 \begin {gather*} \frac {6 \sqrt [6]{a+b x} \left (-91 d^3 (a+b x)^3+399 b d^2 (a+b x)^2 (c+d x)-741 b^2 d (a+b x) (c+d x)^2+1729 b^3 (c+d x)^3\right )}{1729 (b c-a d)^4 (c+d x)^{19/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 171, normalized size = 1.26
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (-1296 b^{3} x^{3} d^{3}+216 d^{3} a \,x^{2} b^{2}-4104 b^{3} c \,d^{2} x^{2}-126 a^{2} b \,d^{3} x +684 a \,b^{2} c \,d^{2} x -4446 b^{3} c^{2} d x +91 a^{3} d^{3}-399 a^{2} b c \,d^{2}+741 a \,b^{2} c^{2} d -1729 b^{3} c^{3}\right )}{1729 \left (d x +c \right )^{\frac {19}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 420 vs.
\(2 (112) = 224\).
time = 1.18, size = 420, normalized size = 3.09 \begin {gather*} \frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 1729 \, b^{3} c^{3} - 741 \, a b^{2} c^{2} d + 399 \, a^{2} b c d^{2} - 91 \, a^{3} d^{3} + 216 \, {\left (19 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (247 \, b^{3} c^{2} d - 38 \, a b^{2} c d^{2} + 7 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{1729 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} + {\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{25/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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