\(\int (d+e x)^2 (a+c x^2)^3 \, dx\) [476]

Optimal result

Integrand size = 17, antiderivative size = 104 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=a^3 d^2 x+\frac {1}{3} a^2 \left (3 c d^2+a e^2\right ) x^3+\frac {3}{5} a c \left (c d^2+a e^2\right ) x^5+\frac {1}{7} c^2 \left (c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 e^2 x^9+\frac {d e \left (a+c x^2\right )^4}{4 c} \]

[Out]

a^3*d^2*x+1/3*a^2*(a*e^2+3*c*d^2)*x^3+3/5*a*c*(a*e^2+c*d^2)*x^5+1/7*c^2*(3*a*e^2+c*d^2)*x^7+1/9*c^3*e^2*x^9+1/
4*d*e*(c*x^2+a)^4/c

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824} \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=a^3 d^2 x+\frac {1}{3} a^2 x^3 \left (a e^2+3 c d^2\right )+\frac {1}{7} c^2 x^7 \left (3 a e^2+c d^2\right )+\frac {3}{5} a c x^5 \left (a e^2+c d^2\right )+\frac {d e \left (a+c x^2\right )^4}{4 c}+\frac {1}{9} c^3 e^2 x^9 \]

[In]

Int[(d + e*x)^2*(a + c*x^2)^3,x]

[Out]

a^3*d^2*x + (a^2*(3*c*d^2 + a*e^2)*x^3)/3 + (3*a*c*(c*d^2 + a*e^2)*x^5)/5 + (c^2*(c*d^2 + 3*a*e^2)*x^7)/7 + (c
^3*e^2*x^9)/9 + (d*e*(a + c*x^2)^4)/(4*c)

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*m*d^(m - 1)*((a + c*x^2)^(p + 1)/
(2*c*(p + 1))), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \frac {d e \left (a+c x^2\right )^4}{4 c}+\int \left (a+c x^2\right )^3 \left (-2 d e x+(d+e x)^2\right ) \, dx \\ & = \frac {d e \left (a+c x^2\right )^4}{4 c}+\int \left (a^3 d^2+a^2 \left (3 c d^2+a e^2\right ) x^2+3 a c \left (c d^2+a e^2\right ) x^4+c^2 \left (c d^2+3 a e^2\right ) x^6+c^3 e^2 x^8\right ) \, dx \\ & = a^3 d^2 x+\frac {1}{3} a^2 \left (3 c d^2+a e^2\right ) x^3+\frac {3}{5} a c \left (c d^2+a e^2\right ) x^5+\frac {1}{7} c^2 \left (c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 e^2 x^9+\frac {d e \left (a+c x^2\right )^4}{4 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} a^2 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+\frac {1}{35} a c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac {1}{252} c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+a^3 \left (d^2 x+d e x^2+\frac {e^2 x^3}{3}\right ) \]

[In]

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

[Out]

(a^2*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/10 + (a*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (c^3*x^7*(3
6*d^2 + 63*d*e*x + 28*e^2*x^2))/252 + a^3*(d^2*x + d*e*x^2 + (e^2*x^3)/3)

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.22

method	result	size
norman	\(\frac {c^{3} e^{2} x^{9}}{9}+\frac {d e \,c^{3} x^{8}}{4}+\left (\frac {3}{7} e^{2} c^{2} a +\frac {1}{7} c^{3} d^{2}\right ) x^{7}+a \,c^{2} d e \,x^{6}+\left (\frac {3}{5} a^{2} c \,e^{2}+\frac {3}{5} a \,c^{2} d^{2}\right ) x^{5}+\frac {3 d e \,a^{2} c \,x^{4}}{2}+\left (\frac {1}{3} a^{3} e^{2}+c \,a^{2} d^{2}\right ) x^{3}+d e \,a^{3} x^{2}+a^{3} d^{2} x\)	\(127\)
default	\(\frac {c^{3} e^{2} x^{9}}{9}+\frac {d e \,c^{3} x^{8}}{4}+\frac {\left (3 e^{2} c^{2} a +c^{3} d^{2}\right ) x^{7}}{7}+a \,c^{2} d e \,x^{6}+\frac {\left (3 a^{2} c \,e^{2}+3 a \,c^{2} d^{2}\right ) x^{5}}{5}+\frac {3 d e \,a^{2} c \,x^{4}}{2}+\frac {\left (a^{3} e^{2}+3 c \,a^{2} d^{2}\right ) x^{3}}{3}+d e \,a^{3} x^{2}+a^{3} d^{2} x\)	\(129\)
gosper	\(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {1}{4} d e \,c^{3} x^{8}+\frac {3}{7} x^{7} e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{2}+a \,c^{2} d e \,x^{6}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} a \,c^{2} d^{2} x^{5}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+a^{2} c \,d^{2} x^{3}+d e \,a^{3} x^{2}+a^{3} d^{2} x\)	\(130\)
risch	\(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {1}{4} d e \,c^{3} x^{8}+\frac {3}{7} x^{7} e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{2}+a \,c^{2} d e \,x^{6}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} a \,c^{2} d^{2} x^{5}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+a^{2} c \,d^{2} x^{3}+d e \,a^{3} x^{2}+a^{3} d^{2} x\)	\(130\)
parallelrisch	\(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {1}{4} d e \,c^{3} x^{8}+\frac {3}{7} x^{7} e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{2}+a \,c^{2} d e \,x^{6}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} a \,c^{2} d^{2} x^{5}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+a^{2} c \,d^{2} x^{3}+d e \,a^{3} x^{2}+a^{3} d^{2} x\)	\(130\)

[In]

int((e*x+d)^2*(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, c^{3} d e x^{8} + a c^{2} d e x^{6} + \frac {3}{2} \, a^{2} c d e x^{4} + \frac {1}{7} \, {\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{7} + a^{3} d e x^{2} + a^{3} d^{2} x + \frac {3}{5} \, {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c d^{2} + a^{3} e^{2}\right )} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.34 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=a^{3} d^{2} x + a^{3} d e x^{2} + \frac {3 a^{2} c d e x^{4}}{2} + a c^{2} d e x^{6} + \frac {c^{3} d e x^{8}}{4} + \frac {c^{3} e^{2} x^{9}}{9} + x^{7} \cdot \left (\frac {3 a c^{2} e^{2}}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c e^{2}}{5} + \frac {3 a c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{3} e^{2}}{3} + a^{2} c d^{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, c^{3} d e x^{8} + a c^{2} d e x^{6} + \frac {3}{2} \, a^{2} c d e x^{4} + \frac {1}{7} \, {\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{7} + a^{3} d e x^{2} + a^{3} d^{2} x + \frac {3}{5} \, {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c d^{2} + a^{3} e^{2}\right )} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.24 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, c^{3} d e x^{8} + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {3}{7} \, a c^{2} e^{2} x^{7} + a c^{2} d e x^{6} + \frac {3}{5} \, a c^{2} d^{2} x^{5} + \frac {3}{5} \, a^{2} c e^{2} x^{5} + \frac {3}{2} \, a^{2} c d e x^{4} + a^{2} c d^{2} x^{3} + \frac {1}{3} \, a^{3} e^{2} x^{3} + a^{3} d e x^{2} + a^{3} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \, dx=x^3\,\left (\frac {a^3\,e^2}{3}+c\,a^2\,d^2\right )+x^7\,\left (\frac {c^3\,d^2}{7}+\frac {3\,a\,c^2\,e^2}{7}\right )+a^3\,d^2\,x+\frac {c^3\,e^2\,x^9}{9}+\frac {3\,a\,c\,x^5\,\left (c\,d^2+a\,e^2\right )}{5}+a^3\,d\,e\,x^2+\frac {c^3\,d\,e\,x^8}{4}+\frac {3\,a^2\,c\,d\,e\,x^4}{2}+a\,c^2\,d\,e\,x^6 \]

[In]

int((a + c*x^2)^3*(d + e*x)^2,x)

[Out]

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