Optimal. Leaf size=35 \[ \frac {x^5}{6 a (a+b x)^6}+\frac {x^5}{30 a^2 (a+b x)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37}
\begin {gather*} \frac {x^5}{30 a^2 (a+b x)^5}+\frac {x^5}{6 a (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x)^7} \, dx &=\frac {x^5}{6 a (a+b x)^6}+\frac {\int \frac {x^4}{(a+b x)^6} \, dx}{6 a}\\ &=\frac {x^5}{6 a (a+b x)^6}+\frac {x^5}{30 a^2 (a+b x)^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 53, normalized size = 1.51 \begin {gather*} -\frac {a^4+6 a^3 b x+15 a^2 b^2 x^2+20 a b^3 x^3+15 b^4 x^4}{30 b^5 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs.
\(2(31)=62\).
time = 0.08, size = 72, normalized size = 2.06
method | result | size |
gosper | \(-\frac {15 b^{4} x^{4}+20 a \,b^{3} x^{3}+15 a^{2} b^{2} x^{2}+6 a^{3} b x +a^{4}}{30 \left (b x +a \right )^{6} b^{5}}\) | \(52\) |
norman | \(\frac {-\frac {x^{4}}{2 b}-\frac {2 a \,x^{3}}{3 b^{2}}-\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a^{3} x}{5 b^{4}}-\frac {a^{4}}{30 b^{5}}}{\left (b x +a \right )^{6}}\) | \(55\) |
risch | \(\frac {-\frac {x^{4}}{2 b}-\frac {2 a \,x^{3}}{3 b^{2}}-\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a^{3} x}{5 b^{4}}-\frac {a^{4}}{30 b^{5}}}{\left (b x +a \right )^{6}}\) | \(55\) |
default | \(-\frac {3 a^{2}}{2 b^{5} \left (b x +a \right )^{4}}-\frac {1}{2 b^{5} \left (b x +a \right )^{2}}-\frac {a^{4}}{6 b^{5} \left (b x +a \right )^{6}}+\frac {4 a^{3}}{5 b^{5} \left (b x +a \right )^{5}}+\frac {4 a}{3 b^{5} \left (b x +a \right )^{3}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs.
\(2 (31) = 62\).
time = 0.30, size = 109, normalized size = 3.11 \begin {gather*} -\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs.
\(2 (31) = 62\).
time = 0.53, size = 109, normalized size = 3.11 \begin {gather*} -\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs.
\(2 (27) = 54\).
time = 0.22, size = 116, normalized size = 3.31 \begin {gather*} \frac {- a^{4} - 6 a^{3} b x - 15 a^{2} b^{2} x^{2} - 20 a b^{3} x^{3} - 15 b^{4} x^{4}}{30 a^{6} b^{5} + 180 a^{5} b^{6} x + 450 a^{4} b^{7} x^{2} + 600 a^{3} b^{8} x^{3} + 450 a^{2} b^{9} x^{4} + 180 a b^{10} x^{5} + 30 b^{11} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.45, size = 51, normalized size = 1.46 \begin {gather*} -\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b x + a\right )}^{6} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 22, normalized size = 0.63 \begin {gather*} \frac {x^5\,\left (6\,a+b\,x\right )}{30\,a^2\,{\left (a+b\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________