Optimal. Leaf size=20 \[ \frac {1}{21 \left (a-b x^3-c x^6\right )^7} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1482, 643}
\begin {gather*} \frac {1}{21 \left (a-b x^3-c x^6\right )^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 643
Rule 1482
Rubi steps
\begin {align*} \int \frac {x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {b+2 c x}{\left (-a+b x+c x^2\right )^8} \, dx,x,x^3\right )\\ &=\frac {1}{21 \left (a-b x^3-c x^6\right )^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{21 \left (-a+b x^3+c x^6\right )^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 19, normalized size = 0.95
method | result | size |
gosper | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
default | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
risch | \(\frac {1}{21 \left (-c \,x^{6}-b \,x^{3}+a \right )^{7}}\) | \(19\) |
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. 356 vs.
\(2 (18) = 36\).
time = 0.42, size = 356, normalized size = 17.80 \begin {gather*} -\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\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. 356 vs.
\(2 (18) = 36\).
time = 0.41, size = 356, normalized size = 17.80 \begin {gather*} -\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\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. 360 vs.
\(2 (15) = 30\).
time = 15.97, size = 360, normalized size = 18.00 \begin {gather*} - \frac {1}{- 21 a^{7} + 147 a^{6} b x^{3} + 147 b c^{6} x^{39} + 21 c^{7} x^{42} + x^{36} \left (- 147 a c^{6} + 441 b^{2} c^{5}\right ) + x^{33} \left (- 882 a b c^{5} + 735 b^{3} c^{4}\right ) + x^{30} \cdot \left (441 a^{2} c^{5} - 2205 a b^{2} c^{4} + 735 b^{4} c^{3}\right ) + x^{27} \cdot \left (2205 a^{2} b c^{4} - 2940 a b^{3} c^{3} + 441 b^{5} c^{2}\right ) + x^{24} \left (- 735 a^{3} c^{4} + 4410 a^{2} b^{2} c^{3} - 2205 a b^{4} c^{2} + 147 b^{6} c\right ) + x^{21} \left (- 2940 a^{3} b c^{3} + 4410 a^{2} b^{3} c^{2} - 882 a b^{5} c + 21 b^{7}\right ) + x^{18} \cdot \left (735 a^{4} c^{3} - 4410 a^{3} b^{2} c^{2} + 2205 a^{2} b^{4} c - 147 a b^{6}\right ) + x^{15} \cdot \left (2205 a^{4} b c^{2} - 2940 a^{3} b^{3} c + 441 a^{2} b^{5}\right ) + x^{12} \left (- 441 a^{5} c^{2} + 2205 a^{4} b^{2} c - 735 a^{3} b^{4}\right ) + x^{9} \left (- 882 a^{5} b c + 735 a^{4} b^{3}\right ) + x^{6} \cdot \left (147 a^{6} c - 441 a^{5} b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.78, size = 18, normalized size = 0.90 \begin {gather*} -\frac {1}{21 \, {\left (c x^{6} + b x^{3} - a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 16.60, size = 360, normalized size = 18.00 \begin {gather*} -\frac {1}{21\,\left (x^{15}\,\left (105\,a^4\,b\,c^2-140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{27}\,\left (105\,a^2\,b\,c^4-140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{21}\,\left (-140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-42\,a\,b^5\,c+b^7\right )+x^9\,\left (35\,a^4\,b^3-42\,a^5\,b\,c\right )+x^{33}\,\left (35\,b^3\,c^4-42\,a\,b\,c^5\right )-x^{12}\,\left (21\,a^5\,c^2-105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{30}\,\left (21\,a^2\,c^5-105\,a\,b^2\,c^4+35\,b^4\,c^3\right )-a^7-x^{18}\,\left (-35\,a^4\,c^3+210\,a^3\,b^2\,c^2-105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^{24}\,\left (-35\,a^3\,c^4+210\,a^2\,b^2\,c^3-105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{42}+x^6\,\left (7\,a^6\,c-21\,a^5\,b^2\right )-x^{36}\,\left (7\,a\,c^6-21\,b^2\,c^5\right )+7\,a^6\,b\,x^3+7\,b\,c^6\,x^{39}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________