### 3.473 $$\int (d+e x)^5 (a+c x^2)^3 \, dx$$

Optimal. Leaf size=190 $\frac{3 c^2 (d+e x)^{10} \left (a e^2+5 c d^2\right )}{10 e^7}-\frac{4 c^2 d (d+e x)^9 \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{8 e^7}-\frac{6 c d (d+e x)^7 \left (a e^2+c d^2\right )^2}{7 e^7}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^3}{6 e^7}+\frac{c^3 (d+e x)^{12}}{12 e^7}-\frac{6 c^3 d (d+e x)^{11}}{11 e^7}$

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^6)/(6*e^7) - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^7)/(7*e^7) + (3*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^8)/(8*e^7) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^7) + (3*c^2*(5*c*d^2
+ a*e^2)*(d + e*x)^10)/(10*e^7) - (6*c^3*d*(d + e*x)^11)/(11*e^7) + (c^3*(d + e*x)^12)/(12*e^7)

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Rubi [A]  time = 0.260904, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $\frac{3 c^2 (d+e x)^{10} \left (a e^2+5 c d^2\right )}{10 e^7}-\frac{4 c^2 d (d+e x)^9 \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{8 e^7}-\frac{6 c d (d+e x)^7 \left (a e^2+c d^2\right )^2}{7 e^7}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^3}{6 e^7}+\frac{c^3 (d+e x)^{12}}{12 e^7}-\frac{6 c^3 d (d+e x)^{11}}{11 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^6)/(6*e^7) - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^7)/(7*e^7) + (3*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^8)/(8*e^7) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^7) + (3*c^2*(5*c*d^2
+ a*e^2)*(d + e*x)^10)/(10*e^7) - (6*c^3*d*(d + e*x)^11)/(11*e^7) + (c^3*(d + e*x)^12)/(12*e^7)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 (d+e x)^5}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^6}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^7}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^8}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^9}{e^6}-\frac{6 c^3 d (d+e x)^{10}}{e^6}+\frac{c^3 (d+e x)^{11}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^3 (d+e x)^6}{6 e^7}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^7}{7 e^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^8}{8 e^7}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^9}{9 e^7}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{10}}{10 e^7}-\frac{6 c^3 d (d+e x)^{11}}{11 e^7}+\frac{c^3 (d+e x)^{12}}{12 e^7}\\ \end{align*}

Mathematica [A]  time = 0.0546938, size = 252, normalized size = 1.33 $a^2 c \left (5 d^2 e^3 x^6+6 d^3 e^2 x^5+\frac{15}{4} d^4 e x^4+d^5 x^3+\frac{15}{7} d e^4 x^7+\frac{3 e^5 x^8}{8}\right )+\frac{1}{6} a^3 x \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+\frac{1}{420} a c^2 x^5 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+\frac{c^3 x^7 \left (6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+3465 d^4 e x+792 d^5+2520 d e^4 x^4+462 e^5 x^5\right )}{5544}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

(a^3*x*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5))/6 + (a*c^2*x^5*(252*d^5
+ 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5))/420 + (c^3*x^7*(792*d^5
+ 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5))/5544 + a^2*c*(d^5*x^3 +
(15*d^4*e*x^4)/4 + 6*d^3*e^2*x^5 + 5*d^2*e^3*x^6 + (15*d*e^4*x^7)/7 + (3*e^5*x^8)/8)

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Maple [A]  time = 0.043, size = 293, normalized size = 1.5 \begin{align*}{\frac{{e}^{5}{c}^{3}{x}^{12}}{12}}+{\frac{5\,d{e}^{4}{c}^{3}{x}^{11}}{11}}+{\frac{ \left ( 3\,{e}^{5}a{c}^{2}+10\,{d}^{2}{e}^{3}{c}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,d{e}^{4}a{c}^{2}+10\,{d}^{3}{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{5}{a}^{2}c+30\,{d}^{2}{e}^{3}a{c}^{2}+5\,{d}^{4}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,d{e}^{4}{a}^{2}c+30\,{d}^{3}{e}^{2}a{c}^{2}+{d}^{5}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}{e}^{5}+30\,{d}^{2}{e}^{3}{a}^{2}c+15\,{d}^{4}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{3}+30\,{d}^{3}{e}^{2}{a}^{2}c+3\,{d}^{5}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{3}+15\,{d}^{4}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{3}+3\,{d}^{5}{a}^{2}c \right ){x}^{3}}{3}}+{\frac{5\,{d}^{4}e{a}^{3}{x}^{2}}{2}}+{d}^{5}{a}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(c*x^2+a)^3,x)

[Out]

1/12*e^5*c^3*x^12+5/11*d*e^4*c^3*x^11+1/10*(3*a*c^2*e^5+10*c^3*d^2*e^3)*x^10+1/9*(15*a*c^2*d*e^4+10*c^3*d^3*e^
2)*x^9+1/8*(3*a^2*c*e^5+30*a*c^2*d^2*e^3+5*c^3*d^4*e)*x^8+1/7*(15*a^2*c*d*e^4+30*a*c^2*d^3*e^2+c^3*d^5)*x^7+1/
6*(a^3*e^5+30*a^2*c*d^2*e^3+15*a*c^2*d^4*e)*x^6+1/5*(5*a^3*d*e^4+30*a^2*c*d^3*e^2+3*a*c^2*d^5)*x^5+1/4*(10*a^3
*d^2*e^3+15*a^2*c*d^4*e)*x^4+1/3*(10*a^3*d^3*e^2+3*a^2*c*d^5)*x^3+5/2*d^4*e*a^3*x^2+d^5*a^3*x

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Maxima [A]  time = 1.38474, size = 394, normalized size = 2.07 \begin{align*} \frac{1}{12} \, c^{3} e^{5} x^{12} + \frac{5}{11} \, c^{3} d e^{4} x^{11} + \frac{1}{10} \,{\left (10 \, c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{10} + \frac{5}{2} \, a^{3} d^{4} e x^{2} + \frac{5}{9} \,{\left (2 \, c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{9} + a^{3} d^{5} x + \frac{1}{8} \,{\left (5 \, c^{3} d^{4} e + 30 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{5} + 30 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (15 \, a c^{2} d^{4} e + 30 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, a c^{2} d^{5} + 30 \, a^{2} c d^{3} e^{2} + 5 \, a^{3} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (3 \, a^{2} c d^{4} e + 2 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a^{2} c d^{5} + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/12*c^3*e^5*x^12 + 5/11*c^3*d*e^4*x^11 + 1/10*(10*c^3*d^2*e^3 + 3*a*c^2*e^5)*x^10 + 5/2*a^3*d^4*e*x^2 + 5/9*(
2*c^3*d^3*e^2 + 3*a*c^2*d*e^4)*x^9 + a^3*d^5*x + 1/8*(5*c^3*d^4*e + 30*a*c^2*d^2*e^3 + 3*a^2*c*e^5)*x^8 + 1/7*
(c^3*d^5 + 30*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*x^7 + 1/6*(15*a*c^2*d^4*e + 30*a^2*c*d^2*e^3 + a^3*e^5)*x^6 + 1/
5*(3*a*c^2*d^5 + 30*a^2*c*d^3*e^2 + 5*a^3*d*e^4)*x^5 + 5/4*(3*a^2*c*d^4*e + 2*a^3*d^2*e^3)*x^4 + 1/3*(3*a^2*c*
d^5 + 10*a^3*d^3*e^2)*x^3

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Fricas [A]  time = 1.62817, size = 667, normalized size = 3.51 \begin{align*} \frac{1}{12} x^{12} e^{5} c^{3} + \frac{5}{11} x^{11} e^{4} d c^{3} + x^{10} e^{3} d^{2} c^{3} + \frac{3}{10} x^{10} e^{5} c^{2} a + \frac{10}{9} x^{9} e^{2} d^{3} c^{3} + \frac{5}{3} x^{9} e^{4} d c^{2} a + \frac{5}{8} x^{8} e d^{4} c^{3} + \frac{15}{4} x^{8} e^{3} d^{2} c^{2} a + \frac{3}{8} x^{8} e^{5} c a^{2} + \frac{1}{7} x^{7} d^{5} c^{3} + \frac{30}{7} x^{7} e^{2} d^{3} c^{2} a + \frac{15}{7} x^{7} e^{4} d c a^{2} + \frac{5}{2} x^{6} e d^{4} c^{2} a + 5 x^{6} e^{3} d^{2} c a^{2} + \frac{1}{6} x^{6} e^{5} a^{3} + \frac{3}{5} x^{5} d^{5} c^{2} a + 6 x^{5} e^{2} d^{3} c a^{2} + x^{5} e^{4} d a^{3} + \frac{15}{4} x^{4} e d^{4} c a^{2} + \frac{5}{2} x^{4} e^{3} d^{2} a^{3} + x^{3} d^{5} c a^{2} + \frac{10}{3} x^{3} e^{2} d^{3} a^{3} + \frac{5}{2} x^{2} e d^{4} a^{3} + x d^{5} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^5*c^3 + 5/11*x^11*e^4*d*c^3 + x^10*e^3*d^2*c^3 + 3/10*x^10*e^5*c^2*a + 10/9*x^9*e^2*d^3*c^3 + 5/3*
x^9*e^4*d*c^2*a + 5/8*x^8*e*d^4*c^3 + 15/4*x^8*e^3*d^2*c^2*a + 3/8*x^8*e^5*c*a^2 + 1/7*x^7*d^5*c^3 + 30/7*x^7*
e^2*d^3*c^2*a + 15/7*x^7*e^4*d*c*a^2 + 5/2*x^6*e*d^4*c^2*a + 5*x^6*e^3*d^2*c*a^2 + 1/6*x^6*e^5*a^3 + 3/5*x^5*d
^5*c^2*a + 6*x^5*e^2*d^3*c*a^2 + x^5*e^4*d*a^3 + 15/4*x^4*e*d^4*c*a^2 + 5/2*x^4*e^3*d^2*a^3 + x^3*d^5*c*a^2 +
10/3*x^3*e^2*d^3*a^3 + 5/2*x^2*e*d^4*a^3 + x*d^5*a^3

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Sympy [A]  time = 0.124749, size = 321, normalized size = 1.69 \begin{align*} a^{3} d^{5} x + \frac{5 a^{3} d^{4} e x^{2}}{2} + \frac{5 c^{3} d e^{4} x^{11}}{11} + \frac{c^{3} e^{5} x^{12}}{12} + x^{10} \left (\frac{3 a c^{2} e^{5}}{10} + c^{3} d^{2} e^{3}\right ) + x^{9} \left (\frac{5 a c^{2} d e^{4}}{3} + \frac{10 c^{3} d^{3} e^{2}}{9}\right ) + x^{8} \left (\frac{3 a^{2} c e^{5}}{8} + \frac{15 a c^{2} d^{2} e^{3}}{4} + \frac{5 c^{3} d^{4} e}{8}\right ) + x^{7} \left (\frac{15 a^{2} c d e^{4}}{7} + \frac{30 a c^{2} d^{3} e^{2}}{7} + \frac{c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{5}}{6} + 5 a^{2} c d^{2} e^{3} + \frac{5 a c^{2} d^{4} e}{2}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} c d^{3} e^{2} + \frac{3 a c^{2} d^{5}}{5}\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{3}}{2} + \frac{15 a^{2} c d^{4} e}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{2}}{3} + a^{2} c d^{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(c*x**2+a)**3,x)

[Out]

a**3*d**5*x + 5*a**3*d**4*e*x**2/2 + 5*c**3*d*e**4*x**11/11 + c**3*e**5*x**12/12 + x**10*(3*a*c**2*e**5/10 + c
**3*d**2*e**3) + x**9*(5*a*c**2*d*e**4/3 + 10*c**3*d**3*e**2/9) + x**8*(3*a**2*c*e**5/8 + 15*a*c**2*d**2*e**3/
4 + 5*c**3*d**4*e/8) + x**7*(15*a**2*c*d*e**4/7 + 30*a*c**2*d**3*e**2/7 + c**3*d**5/7) + x**6*(a**3*e**5/6 + 5
*a**2*c*d**2*e**3 + 5*a*c**2*d**4*e/2) + x**5*(a**3*d*e**4 + 6*a**2*c*d**3*e**2 + 3*a*c**2*d**5/5) + x**4*(5*a
**3*d**2*e**3/2 + 15*a**2*c*d**4*e/4) + x**3*(10*a**3*d**3*e**2/3 + a**2*c*d**5)

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Giac [A]  time = 1.33593, size = 393, normalized size = 2.07 \begin{align*} \frac{1}{12} \, c^{3} x^{12} e^{5} + \frac{5}{11} \, c^{3} d x^{11} e^{4} + c^{3} d^{2} x^{10} e^{3} + \frac{10}{9} \, c^{3} d^{3} x^{9} e^{2} + \frac{5}{8} \, c^{3} d^{4} x^{8} e + \frac{1}{7} \, c^{3} d^{5} x^{7} + \frac{3}{10} \, a c^{2} x^{10} e^{5} + \frac{5}{3} \, a c^{2} d x^{9} e^{4} + \frac{15}{4} \, a c^{2} d^{2} x^{8} e^{3} + \frac{30}{7} \, a c^{2} d^{3} x^{7} e^{2} + \frac{5}{2} \, a c^{2} d^{4} x^{6} e + \frac{3}{5} \, a c^{2} d^{5} x^{5} + \frac{3}{8} \, a^{2} c x^{8} e^{5} + \frac{15}{7} \, a^{2} c d x^{7} e^{4} + 5 \, a^{2} c d^{2} x^{6} e^{3} + 6 \, a^{2} c d^{3} x^{5} e^{2} + \frac{15}{4} \, a^{2} c d^{4} x^{4} e + a^{2} c d^{5} x^{3} + \frac{1}{6} \, a^{3} x^{6} e^{5} + a^{3} d x^{5} e^{4} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{3} d^{4} x^{2} e + a^{3} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/12*c^3*x^12*e^5 + 5/11*c^3*d*x^11*e^4 + c^3*d^2*x^10*e^3 + 10/9*c^3*d^3*x^9*e^2 + 5/8*c^3*d^4*x^8*e + 1/7*c^
3*d^5*x^7 + 3/10*a*c^2*x^10*e^5 + 5/3*a*c^2*d*x^9*e^4 + 15/4*a*c^2*d^2*x^8*e^3 + 30/7*a*c^2*d^3*x^7*e^2 + 5/2*
a*c^2*d^4*x^6*e + 3/5*a*c^2*d^5*x^5 + 3/8*a^2*c*x^8*e^5 + 15/7*a^2*c*d*x^7*e^4 + 5*a^2*c*d^2*x^6*e^3 + 6*a^2*c
*d^3*x^5*e^2 + 15/4*a^2*c*d^4*x^4*e + a^2*c*d^5*x^3 + 1/6*a^3*x^6*e^5 + a^3*d*x^5*e^4 + 5/2*a^3*d^2*x^4*e^3 +
10/3*a^3*d^3*x^3*e^2 + 5/2*a^3*d^4*x^2*e + a^3*d^5*x