Optimal. Leaf size=65 \[ \frac {(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac {2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45}
\begin {gather*} \frac {2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac {(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 640
Rubi steps
\begin {align*} \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^5 (c+d x)^2 \, dx\\ &=\int \left (\frac {(b c-a d)^2 (a+b x)^5}{b^2}+\frac {2 d (b c-a d) (a+b x)^6}{b^2}+\frac {d^2 (a+b x)^7}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac {2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac {d^2 (a+b x)^8}{8 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(189\) vs. \(2(65)=130\).
time = 0.02, size = 189, normalized size = 2.91 \begin {gather*} a^5 c^2 x+\frac {1}{2} a^4 c (5 b c+2 a d) x^2+\frac {1}{3} a^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^3+\frac {5}{4} a^2 b \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+a b^2 \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^5+\frac {1}{6} b^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^6+\frac {1}{7} b^4 d (2 b c+5 a d) x^7+\frac {1}{8} b^5 d^2 x^8 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs.
\(2(59)=118\).
time = 0.70, size = 315, normalized size = 4.85
method | result | size |
norman | \(\frac {b^{5} d^{2} x^{8}}{8}+\left (\frac {5}{7} a \,b^{4} d^{2}+\frac {2}{7} c d \,b^{5}\right ) x^{7}+\left (\frac {5}{3} a^{2} b^{3} d^{2}+\frac {5}{3} a \,b^{4} c d +\frac {1}{6} b^{5} c^{2}\right ) x^{6}+\left (2 b^{2} d^{2} a^{3}+4 a^{2} b^{3} c d +a \,b^{4} c^{2}\right ) x^{5}+\left (\frac {5}{4} d^{2} a^{4} b +5 c d \,a^{3} b^{2}+\frac {5}{2} a^{2} b^{3} c^{2}\right ) x^{4}+\left (\frac {1}{3} d^{2} a^{5}+\frac {10}{3} c d \,a^{4} b +\frac {10}{3} a^{3} b^{2} c^{2}\right ) x^{3}+\left (c d \,a^{5}+\frac {5}{2} c^{2} a^{4} b \right ) x^{2}+c^{2} a^{5} x\) | \(195\) |
risch | \(\frac {1}{8} b^{5} d^{2} x^{8}+\frac {5}{7} x^{7} a \,b^{4} d^{2}+\frac {2}{7} x^{7} c d \,b^{5}+\frac {5}{3} x^{6} a^{2} b^{3} d^{2}+\frac {5}{3} x^{6} a \,b^{4} c d +\frac {1}{6} x^{6} b^{5} c^{2}+2 a^{3} b^{2} d^{2} x^{5}+4 a^{2} b^{3} c d \,x^{5}+a \,b^{4} c^{2} x^{5}+\frac {5}{4} x^{4} d^{2} a^{4} b +5 x^{4} c d \,a^{3} b^{2}+\frac {5}{2} x^{4} a^{2} b^{3} c^{2}+\frac {1}{3} x^{3} d^{2} a^{5}+\frac {10}{3} x^{3} c d \,a^{4} b +\frac {10}{3} x^{3} a^{3} b^{2} c^{2}+x^{2} c d \,a^{5}+\frac {5}{2} x^{2} c^{2} a^{4} b +c^{2} a^{5} x\) | \(213\) |
gosper | \(\frac {x \left (21 b^{5} d^{2} x^{7}+120 x^{6} a \,b^{4} d^{2}+48 x^{6} c d \,b^{5}+280 x^{5} a^{2} b^{3} d^{2}+280 x^{5} a \,b^{4} c d +28 x^{5} b^{5} c^{2}+336 a^{3} b^{2} d^{2} x^{4}+672 a^{2} b^{3} c d \,x^{4}+168 a \,b^{4} c^{2} x^{4}+210 x^{3} d^{2} a^{4} b +840 x^{3} c d \,a^{3} b^{2}+420 x^{3} a^{2} b^{3} c^{2}+56 x^{2} d^{2} a^{5}+560 x^{2} c d \,a^{4} b +560 x^{2} a^{3} b^{2} c^{2}+168 x c d \,a^{5}+420 x \,c^{2} a^{4} b +168 c^{2} a^{5}\right )}{168}\) | \(214\) |
default | \(\frac {b^{5} d^{2} x^{8}}{8}+\frac {\left (3 a \,b^{4} d^{2}+2 b^{4} d \left (a d +b c \right )\right ) x^{7}}{7}+\frac {\left (3 a^{2} b^{3} d^{2}+6 a \,b^{3} d \left (a d +b c \right )+b^{3} \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{6}}{6}+\frac {\left (b^{2} d^{2} a^{3}+6 a^{2} b^{2} d \left (a d +b c \right )+3 a \,b^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 b^{3} a c \left (a d +b c \right )\right ) x^{5}}{5}+\frac {\left (2 a^{3} b d \left (a d +b c \right )+3 a^{2} b \left (\left (a d +b c \right )^{2}+2 a b c d \right )+6 a^{2} b^{2} c \left (a d +b c \right )+a^{2} b^{3} c^{2}\right ) x^{4}}{4}+\frac {\left (a^{3} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+6 a^{3} b c \left (a d +b c \right )+3 a^{3} b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{4} c \left (a d +b c \right )+3 c^{2} a^{4} b \right ) x^{2}}{2}+c^{2} a^{5} x\) | \(315\) |
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. 197 vs.
\(2 (59) = 118\).
time = 0.31, size = 197, normalized size = 3.03 \begin {gather*} \frac {1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} + {\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \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. 197 vs.
\(2 (59) = 118\).
time = 2.48, size = 197, normalized size = 3.03 \begin {gather*} \frac {1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} + {\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \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. 218 vs.
\(2 (56) = 112\).
time = 0.03, size = 218, normalized size = 3.35 \begin {gather*} a^{5} c^{2} x + \frac {b^{5} d^{2} x^{8}}{8} + x^{7} \cdot \left (\frac {5 a b^{4} d^{2}}{7} + \frac {2 b^{5} c d}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{2} b^{3} d^{2}}{3} + \frac {5 a b^{4} c d}{3} + \frac {b^{5} c^{2}}{6}\right ) + x^{5} \cdot \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \cdot \left (\frac {5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac {5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac {a^{5} d^{2}}{3} + \frac {10 a^{4} b c d}{3} + \frac {10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac {5 a^{4} b c^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs.
\(2 (59) = 118\).
time = 0.75, size = 212, normalized size = 3.26 \begin {gather*} \frac {1}{8} \, b^{5} d^{2} x^{8} + \frac {2}{7} \, b^{5} c d x^{7} + \frac {5}{7} \, a b^{4} d^{2} x^{7} + \frac {1}{6} \, b^{5} c^{2} x^{6} + \frac {5}{3} \, a b^{4} c d x^{6} + \frac {5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac {5}{4} \, a^{4} b d^{2} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac {10}{3} \, a^{4} b c d x^{3} + \frac {1}{3} \, a^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 181, normalized size = 2.78 \begin {gather*} x^3\,\left (\frac {a^5\,d^2}{3}+\frac {10\,a^4\,b\,c\,d}{3}+\frac {10\,a^3\,b^2\,c^2}{3}\right )+x^6\,\left (\frac {5\,a^2\,b^3\,d^2}{3}+\frac {5\,a\,b^4\,c\,d}{3}+\frac {b^5\,c^2}{6}\right )+a^5\,c^2\,x+\frac {b^5\,d^2\,x^8}{8}+\frac {a^4\,c\,x^2\,\left (2\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d\,x^7\,\left (5\,a\,d+2\,b\,c\right )}{7}+\frac {5\,a^2\,b\,x^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )}{4}+a\,b^2\,x^5\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________