### 3.474 $$\int (d+e x)^4 (a+c x^2)^3 \, dx$$

Optimal. Leaf size=188 $\frac{c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{2 e^7}+\frac{3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^7}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^3}{5 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7}-\frac{3 c^3 d (d+e x)^{10}}{5 e^7}$

[Out]

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

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Rubi [A]  time = 0.233264, antiderivative size = 188, 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} $\frac{c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{2 e^7}+\frac{3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^7}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^3}{5 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7}-\frac{3 c^3 d (d+e x)^{10}}{5 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(a + c*x^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.0489864, size = 206, normalized size = 1.1 $a^2 c \left (\frac{18}{5} d^2 e^2 x^5+3 d^3 e x^4+d^4 x^3+2 d e^3 x^6+\frac{3 e^4 x^7}{7}\right )+a^3 \left (2 d^2 e^2 x^3+2 d^3 e x^2+d^4 x+d e^3 x^4+\frac{e^4 x^5}{5}\right )+\frac{1}{210} a c^2 x^5 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+\frac{c^3 x^7 \left (1540 d^2 e^2 x^2+1155 d^3 e x+330 d^4+924 d e^3 x^3+210 e^4 x^4\right )}{2310}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4*(a + c*x^2)^3,x]

[Out]

(a*c^2*x^5*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4))/210 + (c^3*x^7*(330*d^4 + 1
155*d^3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4))/2310 + a^3*(d^4*x + 2*d^3*e*x^2 + 2*d^2*e^2*x^3
+ d*e^3*x^4 + (e^4*x^5)/5) + a^2*c*(d^4*x^3 + 3*d^3*e*x^4 + (18*d^2*e^2*x^5)/5 + 2*d*e^3*x^6 + (3*e^4*x^7)/7)

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Maple [A]  time = 0.041, size = 241, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{e}^{4}{x}^{11}}{11}}+{\frac{2\,d{e}^{3}{c}^{3}{x}^{10}}{5}}+{\frac{ \left ( 3\,{e}^{4}a{c}^{2}+6\,{d}^{2}{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 12\,d{e}^{3}a{c}^{2}+4\,{d}^{3}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{e}^{4}{a}^{2}c+18\,{d}^{2}{e}^{2}a{c}^{2}+{d}^{4}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 12\,d{e}^{3}{a}^{2}c+12\,{d}^{3}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}{a}^{3}+18\,{d}^{2}{e}^{2}{a}^{2}c+3\,{d}^{4}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{3}+12\,{d}^{3}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{3}+3\,{d}^{4}{a}^{2}c \right ){x}^{3}}{3}}+2\,{d}^{3}e{a}^{3}{x}^{2}+{d}^{4}{a}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+a)^3,x)

[Out]

1/11*c^3*e^4*x^11+2/5*d*e^3*c^3*x^10+1/9*(3*a*c^2*e^4+6*c^3*d^2*e^2)*x^9+1/8*(12*a*c^2*d*e^3+4*c^3*d^3*e)*x^8+
1/7*(3*a^2*c*e^4+18*a*c^2*d^2*e^2+c^3*d^4)*x^7+1/6*(12*a^2*c*d*e^3+12*a*c^2*d^3*e)*x^6+1/5*(a^3*e^4+18*a^2*c*d
^2*e^2+3*a*c^2*d^4)*x^5+1/4*(4*a^3*d*e^3+12*a^2*c*d^3*e)*x^4+1/3*(6*a^3*d^2*e^2+3*a^2*c*d^4)*x^3+2*d^3*e*a^3*x
^2+d^4*a^3*x

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Maxima [A]  time = 1.87454, size = 313, normalized size = 1.66 \begin{align*} \frac{1}{11} \, c^{3} e^{4} x^{11} + \frac{2}{5} \, c^{3} d e^{3} x^{10} + \frac{1}{3} \,{\left (2 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{9} + 2 \, a^{3} d^{3} e x^{2} + \frac{1}{2} \,{\left (c^{3} d^{3} e + 3 \, a c^{2} d e^{3}\right )} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (c^{3} d^{4} + 18 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x^{7} + 2 \,{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, a c^{2} d^{4} + 18 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{5} +{\left (3 \, a^{2} c d^{3} e + a^{3} d e^{3}\right )} x^{4} +{\left (a^{2} c d^{4} + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 1/3*(2*c^3*d^2*e^2 + a*c^2*e^4)*x^9 + 2*a^3*d^3*e*x^2 + 1/2*(c^3*d^3*
e + 3*a*c^2*d*e^3)*x^8 + a^3*d^4*x + 1/7*(c^3*d^4 + 18*a*c^2*d^2*e^2 + 3*a^2*c*e^4)*x^7 + 2*(a*c^2*d^3*e + a^2
*c*d*e^3)*x^6 + 1/5*(3*a*c^2*d^4 + 18*a^2*c*d^2*e^2 + a^3*e^4)*x^5 + (3*a^2*c*d^3*e + a^3*d*e^3)*x^4 + (a^2*c*
d^4 + 2*a^3*d^2*e^2)*x^3

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Fricas [A]  time = 1.70956, size = 529, normalized size = 2.81 \begin{align*} \frac{1}{11} x^{11} e^{4} c^{3} + \frac{2}{5} x^{10} e^{3} d c^{3} + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} + \frac{1}{3} x^{9} e^{4} c^{2} a + \frac{1}{2} x^{8} e d^{3} c^{3} + \frac{3}{2} x^{8} e^{3} d c^{2} a + \frac{1}{7} x^{7} d^{4} c^{3} + \frac{18}{7} x^{7} e^{2} d^{2} c^{2} a + \frac{3}{7} x^{7} e^{4} c a^{2} + 2 x^{6} e d^{3} c^{2} a + 2 x^{6} e^{3} d c a^{2} + \frac{3}{5} x^{5} d^{4} c^{2} a + \frac{18}{5} x^{5} e^{2} d^{2} c a^{2} + \frac{1}{5} x^{5} e^{4} a^{3} + 3 x^{4} e d^{3} c a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} c a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/11*x^11*e^4*c^3 + 2/5*x^10*e^3*d*c^3 + 2/3*x^9*e^2*d^2*c^3 + 1/3*x^9*e^4*c^2*a + 1/2*x^8*e*d^3*c^3 + 3/2*x^8
*e^3*d*c^2*a + 1/7*x^7*d^4*c^3 + 18/7*x^7*e^2*d^2*c^2*a + 3/7*x^7*e^4*c*a^2 + 2*x^6*e*d^3*c^2*a + 2*x^6*e^3*d*
c*a^2 + 3/5*x^5*d^4*c^2*a + 18/5*x^5*e^2*d^2*c*a^2 + 1/5*x^5*e^4*a^3 + 3*x^4*e*d^3*c*a^2 + x^4*e^3*d*a^3 + x^3
*d^4*c*a^2 + 2*x^3*e^2*d^2*a^3 + 2*x^2*e*d^3*a^3 + x*d^4*a^3

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Sympy [A]  time = 0.111197, size = 255, normalized size = 1.36 \begin{align*} a^{3} d^{4} x + 2 a^{3} d^{3} e x^{2} + \frac{2 c^{3} d e^{3} x^{10}}{5} + \frac{c^{3} e^{4} x^{11}}{11} + x^{9} \left (\frac{a c^{2} e^{4}}{3} + \frac{2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{3 a c^{2} d e^{3}}{2} + \frac{c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac{3 a^{2} c e^{4}}{7} + \frac{18 a c^{2} d^{2} e^{2}}{7} + \frac{c^{3} d^{4}}{7}\right ) + x^{6} \left (2 a^{2} c d e^{3} + 2 a c^{2} d^{3} e\right ) + x^{5} \left (\frac{a^{3} e^{4}}{5} + \frac{18 a^{2} c d^{2} e^{2}}{5} + \frac{3 a c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + 3 a^{2} c d^{3} e\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + a^{2} c d^{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+a)**3,x)

[Out]

a**3*d**4*x + 2*a**3*d**3*e*x**2 + 2*c**3*d*e**3*x**10/5 + c**3*e**4*x**11/11 + x**9*(a*c**2*e**4/3 + 2*c**3*d
**2*e**2/3) + x**8*(3*a*c**2*d*e**3/2 + c**3*d**3*e/2) + x**7*(3*a**2*c*e**4/7 + 18*a*c**2*d**2*e**2/7 + c**3*
d**4/7) + x**6*(2*a**2*c*d*e**3 + 2*a*c**2*d**3*e) + x**5*(a**3*e**4/5 + 18*a**2*c*d**2*e**2/5 + 3*a*c**2*d**4
/5) + x**4*(a**3*d*e**3 + 3*a**2*c*d**3*e) + x**3*(2*a**3*d**2*e**2 + a**2*c*d**4)

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Giac [A]  time = 1.32047, size = 321, normalized size = 1.71 \begin{align*} \frac{1}{11} \, c^{3} x^{11} e^{4} + \frac{2}{5} \, c^{3} d x^{10} e^{3} + \frac{2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac{1}{2} \, c^{3} d^{3} x^{8} e + \frac{1}{7} \, c^{3} d^{4} x^{7} + \frac{1}{3} \, a c^{2} x^{9} e^{4} + \frac{3}{2} \, a c^{2} d x^{8} e^{3} + \frac{18}{7} \, a c^{2} d^{2} x^{7} e^{2} + 2 \, a c^{2} d^{3} x^{6} e + \frac{3}{5} \, a c^{2} d^{4} x^{5} + \frac{3}{7} \, a^{2} c x^{7} e^{4} + 2 \, a^{2} c d x^{6} e^{3} + \frac{18}{5} \, a^{2} c d^{2} x^{5} e^{2} + 3 \, a^{2} c d^{3} x^{4} e + a^{2} c d^{4} x^{3} + \frac{1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/11*c^3*x^11*e^4 + 2/5*c^3*d*x^10*e^3 + 2/3*c^3*d^2*x^9*e^2 + 1/2*c^3*d^3*x^8*e + 1/7*c^3*d^4*x^7 + 1/3*a*c^2
*x^9*e^4 + 3/2*a*c^2*d*x^8*e^3 + 18/7*a*c^2*d^2*x^7*e^2 + 2*a*c^2*d^3*x^6*e + 3/5*a*c^2*d^4*x^5 + 3/7*a^2*c*x^
7*e^4 + 2*a^2*c*d*x^6*e^3 + 18/5*a^2*c*d^2*x^5*e^2 + 3*a^2*c*d^3*x^4*e + a^2*c*d^4*x^3 + 1/5*a^3*x^5*e^4 + a^3
*d*x^4*e^3 + 2*a^3*d^2*x^3*e^2 + 2*a^3*d^3*x^2*e + a^3*d^4*x