Optimal. Leaf size=119 \[ \frac{2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac{2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac{e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac{(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.251798, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac{2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac{e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac{(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^4 \, dx\\ &=\int \left (\frac{(b d-a e)^4 (a+b x)^6}{b^4}+\frac{4 e (b d-a e)^3 (a+b x)^7}{b^4}+\frac{6 e^2 (b d-a e)^2 (a+b x)^8}{b^4}+\frac{4 e^3 (b d-a e) (a+b x)^9}{b^4}+\frac{e^4 (a+b x)^{10}}{b^4}\right ) \, dx\\ &=\frac{(b d-a e)^4 (a+b x)^7}{7 b^5}+\frac{e (b d-a e)^3 (a+b x)^8}{2 b^5}+\frac{2 e^2 (b d-a e)^2 (a+b x)^9}{3 b^5}+\frac{2 e^3 (b d-a e) (a+b x)^{10}}{5 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5}\\ \end{align*}
Mathematica [B] time = 0.061379, size = 398, normalized size = 3.34 \[ \frac{1}{3} b^4 e^2 x^9 \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+\frac{1}{2} b^3 e x^8 \left (15 a^2 b d e^2+5 a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{7} b^2 x^7 \left (90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4+24 a b^3 d^3 e+b^4 d^4\right )+a b x^6 \left (20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4+10 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{5} a^2 x^5 \left (90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4+80 a b^3 d^3 e+15 b^4 d^4\right )+a^3 d x^4 \left (9 a^2 b d e^2+a^3 e^3+15 a b^2 d^2 e+5 b^3 d^3\right )+a^4 d^2 x^3 \left (2 a^2 e^2+8 a b d e+5 b^2 d^2\right )+a^5 d^3 x^2 (2 a e+3 b d)+a^6 d^4 x+\frac{1}{5} b^5 e^3 x^{10} (3 a e+2 b d)+\frac{1}{11} b^6 e^4 x^{11} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.04, size = 427, normalized size = 3.6 \begin{align*}{\frac{{e}^{4}{b}^{6}{x}^{11}}{11}}+{\frac{ \left ( 6\,{e}^{4}a{b}^{5}+4\,d{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,{e}^{4}{a}^{2}{b}^{4}+24\,d{e}^{3}a{b}^{5}+6\,{d}^{2}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 20\,{e}^{4}{a}^{3}{b}^{3}+60\,d{e}^{3}{a}^{2}{b}^{4}+36\,{d}^{2}{e}^{2}a{b}^{5}+4\,{d}^{3}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,{e}^{4}{a}^{4}{b}^{2}+80\,d{e}^{3}{a}^{3}{b}^{3}+90\,{d}^{2}{e}^{2}{a}^{2}{b}^{4}+24\,{d}^{3}ea{b}^{5}+{d}^{4}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{e}^{4}{a}^{5}b+60\,d{e}^{3}{a}^{4}{b}^{2}+120\,{d}^{2}{e}^{2}{a}^{3}{b}^{3}+60\,{d}^{3}e{a}^{2}{b}^{4}+6\,{d}^{4}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}{a}^{6}+24\,d{e}^{3}{a}^{5}b+90\,{d}^{2}{e}^{2}{a}^{4}{b}^{2}+80\,{d}^{3}e{a}^{3}{b}^{3}+15\,{d}^{4}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{6}+36\,{d}^{2}{e}^{2}{a}^{5}b+60\,{d}^{3}e{a}^{4}{b}^{2}+20\,{d}^{4}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{6}+24\,{d}^{3}e{a}^{5}b+15\,{d}^{4}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{6}+6\,{d}^{4}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{4}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.1732, size = 564, normalized size = 4.74 \begin{align*} \frac{1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac{1}{5} \,{\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac{1}{2} \,{\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} +{\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} +{\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.50303, size = 992, normalized size = 8.34 \begin{align*} \frac{1}{11} x^{11} e^{4} b^{6} + \frac{2}{5} x^{10} e^{3} d b^{6} + \frac{3}{5} x^{10} e^{4} b^{5} a + \frac{2}{3} x^{9} e^{2} d^{2} b^{6} + \frac{8}{3} x^{9} e^{3} d b^{5} a + \frac{5}{3} x^{9} e^{4} b^{4} a^{2} + \frac{1}{2} x^{8} e d^{3} b^{6} + \frac{9}{2} x^{8} e^{2} d^{2} b^{5} a + \frac{15}{2} x^{8} e^{3} d b^{4} a^{2} + \frac{5}{2} x^{8} e^{4} b^{3} a^{3} + \frac{1}{7} x^{7} d^{4} b^{6} + \frac{24}{7} x^{7} e d^{3} b^{5} a + \frac{90}{7} x^{7} e^{2} d^{2} b^{4} a^{2} + \frac{80}{7} x^{7} e^{3} d b^{3} a^{3} + \frac{15}{7} x^{7} e^{4} b^{2} a^{4} + x^{6} d^{4} b^{5} a + 10 x^{6} e d^{3} b^{4} a^{2} + 20 x^{6} e^{2} d^{2} b^{3} a^{3} + 10 x^{6} e^{3} d b^{2} a^{4} + x^{6} e^{4} b a^{5} + 3 x^{5} d^{4} b^{4} a^{2} + 16 x^{5} e d^{3} b^{3} a^{3} + 18 x^{5} e^{2} d^{2} b^{2} a^{4} + \frac{24}{5} x^{5} e^{3} d b a^{5} + \frac{1}{5} x^{5} e^{4} a^{6} + 5 x^{4} d^{4} b^{3} a^{3} + 15 x^{4} e d^{3} b^{2} a^{4} + 9 x^{4} e^{2} d^{2} b a^{5} + x^{4} e^{3} d a^{6} + 5 x^{3} d^{4} b^{2} a^{4} + 8 x^{3} e d^{3} b a^{5} + 2 x^{3} e^{2} d^{2} a^{6} + 3 x^{2} d^{4} b a^{5} + 2 x^{2} e d^{3} a^{6} + x d^{4} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.150309, size = 462, normalized size = 3.88 \begin{align*} a^{6} d^{4} x + \frac{b^{6} e^{4} x^{11}}{11} + x^{10} \left (\frac{3 a b^{5} e^{4}}{5} + \frac{2 b^{6} d e^{3}}{5}\right ) + x^{9} \left (\frac{5 a^{2} b^{4} e^{4}}{3} + \frac{8 a b^{5} d e^{3}}{3} + \frac{2 b^{6} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{5 a^{3} b^{3} e^{4}}{2} + \frac{15 a^{2} b^{4} d e^{3}}{2} + \frac{9 a b^{5} d^{2} e^{2}}{2} + \frac{b^{6} d^{3} e}{2}\right ) + x^{7} \left (\frac{15 a^{4} b^{2} e^{4}}{7} + \frac{80 a^{3} b^{3} d e^{3}}{7} + \frac{90 a^{2} b^{4} d^{2} e^{2}}{7} + \frac{24 a b^{5} d^{3} e}{7} + \frac{b^{6} d^{4}}{7}\right ) + x^{6} \left (a^{5} b e^{4} + 10 a^{4} b^{2} d e^{3} + 20 a^{3} b^{3} d^{2} e^{2} + 10 a^{2} b^{4} d^{3} e + a b^{5} d^{4}\right ) + x^{5} \left (\frac{a^{6} e^{4}}{5} + \frac{24 a^{5} b d e^{3}}{5} + 18 a^{4} b^{2} d^{2} e^{2} + 16 a^{3} b^{3} d^{3} e + 3 a^{2} b^{4} d^{4}\right ) + x^{4} \left (a^{6} d e^{3} + 9 a^{5} b d^{2} e^{2} + 15 a^{4} b^{2} d^{3} e + 5 a^{3} b^{3} d^{4}\right ) + x^{3} \left (2 a^{6} d^{2} e^{2} + 8 a^{5} b d^{3} e + 5 a^{4} b^{2} d^{4}\right ) + x^{2} \left (2 a^{6} d^{3} e + 3 a^{5} b d^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.12307, size = 616, normalized size = 5.18 \begin{align*} \frac{1}{11} \, b^{6} x^{11} e^{4} + \frac{2}{5} \, b^{6} d x^{10} e^{3} + \frac{2}{3} \, b^{6} d^{2} x^{9} e^{2} + \frac{1}{2} \, b^{6} d^{3} x^{8} e + \frac{1}{7} \, b^{6} d^{4} x^{7} + \frac{3}{5} \, a b^{5} x^{10} e^{4} + \frac{8}{3} \, a b^{5} d x^{9} e^{3} + \frac{9}{2} \, a b^{5} d^{2} x^{8} e^{2} + \frac{24}{7} \, a b^{5} d^{3} x^{7} e + a b^{5} d^{4} x^{6} + \frac{5}{3} \, a^{2} b^{4} x^{9} e^{4} + \frac{15}{2} \, a^{2} b^{4} d x^{8} e^{3} + \frac{90}{7} \, a^{2} b^{4} d^{2} x^{7} e^{2} + 10 \, a^{2} b^{4} d^{3} x^{6} e + 3 \, a^{2} b^{4} d^{4} x^{5} + \frac{5}{2} \, a^{3} b^{3} x^{8} e^{4} + \frac{80}{7} \, a^{3} b^{3} d x^{7} e^{3} + 20 \, a^{3} b^{3} d^{2} x^{6} e^{2} + 16 \, a^{3} b^{3} d^{3} x^{5} e + 5 \, a^{3} b^{3} d^{4} x^{4} + \frac{15}{7} \, a^{4} b^{2} x^{7} e^{4} + 10 \, a^{4} b^{2} d x^{6} e^{3} + 18 \, a^{4} b^{2} d^{2} x^{5} e^{2} + 15 \, a^{4} b^{2} d^{3} x^{4} e + 5 \, a^{4} b^{2} d^{4} x^{3} + a^{5} b x^{6} e^{4} + \frac{24}{5} \, a^{5} b d x^{5} e^{3} + 9 \, a^{5} b d^{2} x^{4} e^{2} + 8 \, a^{5} b d^{3} x^{3} e + 3 \, a^{5} b d^{4} x^{2} + \frac{1}{5} \, a^{6} x^{5} e^{4} + a^{6} d x^{4} e^{3} + 2 \, a^{6} d^{2} x^{3} e^{2} + 2 \, a^{6} d^{3} x^{2} e + a^{6} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]