Optimal. Leaf size=65 \[ \frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \]
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Rubi [A]
time = 0.06, 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 {d (a+b x)^6 (b c-a d)}{3 b^3}+\frac {(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^4 (c+d x)^2 \, dx\\ &=\int \left (\frac {(b c-a d)^2 (a+b x)^4}{b^2}+\frac {2 d (b c-a d) (a+b x)^5}{b^2}+\frac {d^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(65)=130\).
time = 0.01, size = 148, normalized size = 2.28 \begin {gather*} a^4 c^2 x+a^3 c (2 b c+a d) x^2+\frac {1}{3} a^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^3+a b \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^4+\frac {1}{5} b^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^5+\frac {1}{3} b^3 d (b c+2 a d) x^6+\frac {1}{7} b^4 d^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs.
\(2(59)=118\).
time = 0.80, size = 231, normalized size = 3.55
method | result | size |
norman | \(\frac {b^{4} d^{2} x^{7}}{7}+\left (\frac {2}{3} d^{2} a \,b^{3}+\frac {1}{3} c d \,b^{4}\right ) x^{6}+\left (\frac {6}{5} a^{2} b^{2} d^{2}+\frac {8}{5} c d a \,b^{3}+\frac {1}{5} b^{4} c^{2}\right ) x^{5}+\left (a^{3} b \,d^{2}+3 a^{2} b^{2} c d +a \,b^{3} c^{2}\right ) x^{4}+\left (\frac {1}{3} d^{2} a^{4}+\frac {8}{3} c d \,a^{3} b +2 a^{2} b^{2} c^{2}\right ) x^{3}+\left (c d \,a^{4}+2 c^{2} a^{3} b \right ) x^{2}+c^{2} a^{4} x\) | \(157\) |
risch | \(\frac {1}{7} b^{4} d^{2} x^{7}+\frac {2}{3} x^{6} d^{2} a \,b^{3}+\frac {1}{3} x^{6} c d \,b^{4}+\frac {6}{5} x^{5} a^{2} b^{2} d^{2}+\frac {8}{5} x^{5} c d a \,b^{3}+\frac {1}{5} x^{5} b^{4} c^{2}+a^{3} b \,d^{2} x^{4}+3 a^{2} b^{2} c d \,x^{4}+a \,b^{3} c^{2} x^{4}+\frac {1}{3} x^{3} d^{2} a^{4}+\frac {8}{3} x^{3} c d \,a^{3} b +2 x^{3} a^{2} b^{2} c^{2}+a^{4} c d \,x^{2}+2 a^{3} b \,c^{2} x^{2}+c^{2} a^{4} x\) | \(171\) |
gosper | \(\frac {x \left (15 b^{4} d^{2} x^{6}+70 x^{5} d^{2} a \,b^{3}+35 x^{5} c d \,b^{4}+126 x^{4} a^{2} b^{2} d^{2}+168 x^{4} c d a \,b^{3}+21 x^{4} b^{4} c^{2}+105 a^{3} b \,d^{2} x^{3}+315 a^{2} b^{2} c d \,x^{3}+105 a \,b^{3} c^{2} x^{3}+35 x^{2} d^{2} a^{4}+280 x^{2} c d \,a^{3} b +210 x^{2} a^{2} b^{2} c^{2}+105 a^{4} c d x +210 a^{3} b \,c^{2} x +105 c^{2} a^{4}\right )}{105}\) | \(173\) |
default | \(\frac {b^{4} d^{2} x^{7}}{7}+\frac {\left (2 d^{2} a \,b^{3}+2 b^{3} d \left (a d +b c \right )\right ) x^{6}}{6}+\frac {\left (a^{2} b^{2} d^{2}+4 a \,b^{2} d \left (a d +b c \right )+b^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{5}}{5}+\frac {\left (2 a^{2} b d \left (a d +b c \right )+2 a b \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 b^{2} a c \left (a d +b c \right )\right ) x^{4}}{4}+\frac {\left (a^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+4 a^{2} b c \left (a d +b c \right )+a^{2} b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} c \left (a d +b c \right )+2 c^{2} a^{3} b \right ) x^{2}}{2}+c^{2} a^{4} x\) | \(231\) |
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. 156 vs.
\(2 (59) = 118\).
time = 0.29, size = 156, normalized size = 2.40 \begin {gather*} \frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \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. 156 vs.
\(2 (59) = 118\).
time = 2.02, size = 156, normalized size = 2.40 \begin {gather*} \frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \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. 168 vs.
\(2 (54) = 108\).
time = 0.03, size = 168, normalized size = 2.58 \begin {gather*} a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \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. 170 vs.
\(2 (59) = 118\).
time = 0.90, size = 170, normalized size = 2.62 \begin {gather*} \frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, b^{4} c d x^{6} + \frac {2}{3} \, a b^{3} d^{2} x^{6} + \frac {1}{5} \, b^{4} c^{2} x^{5} + \frac {8}{5} \, a b^{3} c d x^{5} + \frac {6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac {8}{3} \, a^{3} b c d x^{3} + \frac {1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 144, normalized size = 2.22 \begin {gather*} x^3\,\left (\frac {a^4\,d^2}{3}+\frac {8\,a^3\,b\,c\,d}{3}+2\,a^2\,b^2\,c^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,d^2}{5}+\frac {8\,a\,b^3\,c\,d}{5}+\frac {b^4\,c^2}{5}\right )+a^4\,c^2\,x+\frac {b^4\,d^2\,x^7}{7}+a^3\,c\,x^2\,\left (a\,d+2\,b\,c\right )+\frac {b^3\,d\,x^6\,\left (2\,a\,d+b\,c\right )}{3}+a\,b\,x^4\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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