Optimal. Leaf size=133 \[ \frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6} \]
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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} -\frac {5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac {10 b^3 x (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac {b^5 (c+d x)^3}{3 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{(c+d x)^3} \, dx &=\int \left (\frac {10 b^3 (b c-a d)^2}{d^5}+\frac {(-b c+a d)^5}{d^5 (c+d x)^3}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^2}-\frac {10 b^2 (b c-a d)^3}{d^5 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)}{d^5}+\frac {b^5 (c+d x)^2}{d^5}\right ) \, dx\\ &=\frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 230, normalized size = 1.73 \begin {gather*} \frac {-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+15 a b^4 d \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+b^5 \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )-60 b^2 (b c-a d)^3 (c+d x)^2 \log (c+d x)}{6 d^6 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(290\) vs. \(2(133)=266\).
time = 4.77, size = 288, normalized size = 2.17 \begin {gather*} \frac {-3 a^5 d^5-15 a^4 b c d^4+60 b^2 \text {Log}\left [c+d x\right ] \left (c^2+2 c d x+d^2 x^2\right ) \left (a d-b c\right )^3+90 a^3 b^2 c^2 d^3-150 a^2 b^3 c^3 d^2+105 a b^4 c^4 d-27 b^5 c^5-30 b d x \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 )+6 b^3 d x \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \left (c^2+2 c d x+d^2 x^2\right )+3 b^4 d^2 x^2 \left (5 a d-3 b c\right ) \left (c^2+2 c d x+d^2 x^2\right )+2 b^5 d^3 x^3 \left (c^2+2 c d x+d^2 x^2\right )}{6 d^6 \left (c^2+2 c d x+d^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 254, normalized size = 1.91
method | result | size |
default | \(\frac {b^{3} \left (\frac {1}{3} d^{2} x^{3} b^{2}+\frac {5}{2} a b \,d^{2} x^{2}-\frac {3}{2} b^{2} c d \,x^{2}+10 a^{2} d^{2} x -15 a b c d x +6 b^{2} c^{2} x \right )}{d^{5}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{d^{6} \left (d x +c \right )}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{2 d^{6} \left (d x +c \right )^{2}}\) | \(254\) |
norman | \(\frac {-\frac {a^{5} d^{5}+5 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}+90 a^{2} b^{3} c^{3} d^{2}-90 a \,b^{4} c^{4} d +30 b^{5} c^{5}}{2 d^{6}}+\frac {b^{5} x^{5}}{3 d}-\frac {\left (5 a^{4} b \,d^{4}-20 a^{3} b^{2} c \,d^{3}+60 a^{2} b^{3} c^{2} d^{2}-60 a \,b^{4} c^{3} d +20 b^{5} c^{4}\right ) x}{d^{5}}+\frac {10 b^{3} \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{3}}{3 d^{3}}+\frac {5 b^{4} \left (3 a d -b c \right ) x^{4}}{6 d^{2}}}{\left (d x +c \right )^{2}}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(254\) |
risch | \(\frac {b^{5} x^{3}}{3 d^{3}}+\frac {5 b^{4} a \,x^{2}}{2 d^{3}}-\frac {3 b^{5} c \,x^{2}}{2 d^{4}}+\frac {10 b^{3} a^{2} x}{d^{3}}-\frac {15 b^{4} a c x}{d^{4}}+\frac {6 b^{5} c^{2} x}{d^{5}}+\frac {\left (-5 a^{4} b \,d^{4}+20 a^{3} b^{2} c \,d^{3}-30 a^{2} b^{3} c^{2} d^{2}+20 a \,b^{4} c^{3} d -5 b^{5} c^{4}\right ) x -\frac {a^{5} d^{5}+5 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}+50 a^{2} b^{3} c^{3} d^{2}-35 a \,b^{4} c^{4} d +9 b^{5} c^{5}}{2 d}}{d^{5} \left (d x +c \right )^{2}}+\frac {10 b^{2} \ln \left (d x +c \right ) a^{3}}{d^{3}}-\frac {30 b^{3} \ln \left (d x +c \right ) a^{2} c}{d^{4}}+\frac {30 b^{4} \ln \left (d x +c \right ) a \,c^{2}}{d^{5}}-\frac {10 b^{5} \ln \left (d x +c \right ) c^{3}}{d^{6}}\) | \(279\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs.
\(2 (127) = 254\).
time = 0.27, size = 271, normalized size = 2.04 \begin {gather*} -\frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {2 \, b^{5} d^{2} x^{3} - 3 \, {\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \, {\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \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. 416 vs.
\(2 (127) = 254\).
time = 0.29, size = 416, normalized size = 3.13 \begin {gather*} \frac {2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \, {\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs.
\(2 (121) = 242\).
time = 1.00, size = 258, normalized size = 1.94 \begin {gather*} \frac {b^{5} x^{3}}{3 d^{3}} + \frac {10 b^{2} \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{6}} + x^{2} \cdot \left (\frac {5 a b^{4}}{2 d^{3}} - \frac {3 b^{5} c}{2 d^{4}}\right ) + x \left (\frac {10 a^{2} b^{3}}{d^{3}} - \frac {15 a b^{4} c}{d^{4}} + \frac {6 b^{5} c^{2}}{d^{5}}\right ) + \frac {- a^{5} d^{5} - 5 a^{4} b c d^{4} + 30 a^{3} b^{2} c^{2} d^{3} - 50 a^{2} b^{3} c^{3} d^{2} + 35 a b^{4} c^{4} d - 9 b^{5} c^{5} + x \left (- 10 a^{4} b d^{5} + 40 a^{3} b^{2} c d^{4} - 60 a^{2} b^{3} c^{2} d^{3} + 40 a b^{4} c^{3} d^{2} - 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} \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. 264 vs.
\(2 (127) = 254\).
time = 0.00, size = 287, normalized size = 2.16 \begin {gather*} \frac {\frac {1}{3} x^{3} b^{5} d^{6}-\frac {3}{2} x^{2} b^{5} d^{5} c+\frac {5}{2} x^{2} b^{4} a d^{6}+6 x b^{5} d^{4} c^{2}-15 x b^{4} a d^{5} c+10 x b^{3} a^{2} d^{6}}{d^{9}}+\frac {\frac {1}{2} \left (-9 b^{5} c^{5}+35 b^{4} d c^{4} a-50 b^{3} d^{2} c^{3} a^{2}+30 b^{2} d^{3} c^{2} a^{3}-5 b d^{4} c a^{4}-d^{5} a^{5}+\left (-10 b^{5} d c^{4}+40 b^{4} d^{2} c^{3} a-60 b^{3} d^{3} c^{2} a^{2}+40 b^{2} d^{4} c a^{3}-10 b d^{5} a^{4}\right ) x\right )}{d^{6} \left (x d+c\right )^{2}}+\frac {\left (-10 b^{5} c^{3}+30 b^{4} a d c^{2}-30 b^{3} a^{2} d^{2} c+10 b^{2} a^{3} d^{3}\right ) \ln \left |x d+c\right |}{d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 291, normalized size = 2.19 \begin {gather*} x^2\,\left (\frac {5\,a\,b^4}{2\,d^3}-\frac {3\,b^5\,c}{2\,d^4}\right )-\frac {\frac {a^5\,d^5+5\,a^4\,b\,c\,d^4-30\,a^3\,b^2\,c^2\,d^3+50\,a^2\,b^3\,c^3\,d^2-35\,a\,b^4\,c^4\,d+9\,b^5\,c^5}{2\,d}+x\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{c^2\,d^5+2\,c\,d^6\,x+d^7\,x^2}-x\,\left (\frac {3\,c\,\left (\frac {5\,a\,b^4}{d^3}-\frac {3\,b^5\,c}{d^4}\right )}{d}-\frac {10\,a^2\,b^3}{d^3}+\frac {3\,b^5\,c^2}{d^5}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )}{d^6}+\frac {b^5\,x^3}{3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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