Optimal. Leaf size=58 \[ \frac {b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac {(a+b x)^6}{7 (c+d x)^7 (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac {b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac {(a+b x)^6}{7 (c+d x)^7 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{(c+d x)^8} \, dx &=\frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b \int \frac {(a+b x)^5}{(c+d x)^7} \, dx}{7 (b c-a d)}\\ &=\frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^6}{42 (b c-a d)^2 (c+d x)^6}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 205, normalized size = 3.53 \[ -\frac {6 a^5 d^5+5 a^4 b d^4 (c+7 d x)+4 a^3 b^2 d^3 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a^2 b^3 d^2 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+2 a b^4 d \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )}{42 d^6 (c+d x)^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 326, normalized size = 5.62 \[ -\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 271, normalized size = 4.67 \[ -\frac {21 \, b^{5} d^{5} x^{5} + 35 \, b^{5} c d^{4} x^{4} + 70 \, a b^{4} d^{5} x^{4} + 35 \, b^{5} c^{2} d^{3} x^{3} + 70 \, a b^{4} c d^{4} x^{3} + 105 \, a^{2} b^{3} d^{5} x^{3} + 21 \, b^{5} c^{3} d^{2} x^{2} + 42 \, a b^{4} c^{2} d^{3} x^{2} + 63 \, a^{2} b^{3} c d^{4} x^{2} + 84 \, a^{3} b^{2} d^{5} x^{2} + 7 \, b^{5} c^{4} d x + 14 \, a b^{4} c^{3} d^{2} x + 21 \, a^{2} b^{3} c^{2} d^{3} x + 28 \, a^{3} b^{2} c d^{4} x + 35 \, a^{4} b d^{5} x + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5}}{42 \, {\left (d x + c\right )}^{7} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 265, normalized size = 4.57 \[ -\frac {b^{5}}{2 \left (d x +c \right )^{2} d^{6}}-\frac {5 \left (a d -b c \right ) b^{4}}{3 \left (d x +c \right )^{3} d^{6}}-\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3}}{2 \left (d x +c \right )^{4} d^{6}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2}}{\left (d x +c \right )^{5} d^{6}}-\frac {5 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b}{6 \left (d x +c \right )^{6} d^{6}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{7 \left (d x +c \right )^{7} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.58, size = 326, normalized size = 5.62 \[ -\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 39, normalized size = 0.67 \[ \frac {{\left (a+b\,x\right )}^6\,\left (7\,b\,c-6\,a\,d+b\,d\,x\right )}{42\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 54.91, size = 354, normalized size = 6.10 \[ \frac {- 6 a^{5} d^{5} - 5 a^{4} b c d^{4} - 4 a^{3} b^{2} c^{2} d^{3} - 3 a^{2} b^{3} c^{3} d^{2} - 2 a b^{4} c^{4} d - b^{5} c^{5} - 21 b^{5} d^{5} x^{5} + x^{4} \left (- 70 a b^{4} d^{5} - 35 b^{5} c d^{4}\right ) + x^{3} \left (- 105 a^{2} b^{3} d^{5} - 70 a b^{4} c d^{4} - 35 b^{5} c^{2} d^{3}\right ) + x^{2} \left (- 84 a^{3} b^{2} d^{5} - 63 a^{2} b^{3} c d^{4} - 42 a b^{4} c^{2} d^{3} - 21 b^{5} c^{3} d^{2}\right ) + x \left (- 35 a^{4} b d^{5} - 28 a^{3} b^{2} c d^{4} - 21 a^{2} b^{3} c^{2} d^{3} - 14 a b^{4} c^{3} d^{2} - 7 b^{5} c^{4} d\right )}{42 c^{7} d^{6} + 294 c^{6} d^{7} x + 882 c^{5} d^{8} x^{2} + 1470 c^{4} d^{9} x^{3} + 1470 c^{3} d^{10} x^{4} + 882 c^{2} d^{11} x^{5} + 294 c d^{12} x^{6} + 42 d^{13} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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