\(\int (d+e x) (a+c x^2)^3 \, dx\) [477]

Optimal result

Integrand size = 15, antiderivative size = 56 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {1}{7} c^3 d x^7+\frac {e \left (a+c x^2\right )^4}{8 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200} \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {e \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} c^3 d x^7 \]

[In]

Int[(d + e*x)*(a + c*x^2)^3,x]

[Out]

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 655

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+a^2 c d x^3+\frac {3}{4} a^2 c e x^4+\frac {3}{5} a c^2 d x^5+\frac {1}{2} a c^2 e x^6+\frac {1}{7} c^3 d x^7+\frac {1}{8} c^3 e x^8 \]

[In]

Integrate[(d + e*x)*(a + c*x^2)^3,x]

[Out]

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

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32

method	result	size
gosper	\(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\)	\(74\)
default	\(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\)	\(74\)
norman	\(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\)	\(74\)
risch	\(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\)	\(74\)
parallelrisch	\(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\)	\(74\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^{3} d x + \frac {a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac {3 a^{2} c e x^{4}}{4} + \frac {3 a c^{2} d x^{5}}{5} + \frac {a c^{2} e x^{6}}{2} + \frac {c^{3} d x^{7}}{7} + \frac {c^{3} e x^{8}}{8} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {e\,a^3\,x^2}{2}+d\,a^3\,x+\frac {3\,e\,a^2\,c\,x^4}{4}+d\,a^2\,c\,x^3+\frac {e\,a\,c^2\,x^6}{2}+\frac {3\,d\,a\,c^2\,x^5}{5}+\frac {e\,c^3\,x^8}{8}+\frac {d\,c^3\,x^7}{7} \]

[In]

int((a + c*x^2)^3*(d + e*x),x)

[Out]

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