\(\int (d+e x) (a+c x^2)^4 \, dx\) [494]

Optimal result

Integrand size = 15, antiderivative size = 73 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {1}{9} c^4 d x^9+\frac {e \left (a+c x^2\right )^5}{10 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, 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} \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {e \left (a+c x^2\right )^5}{10 c}+\frac {1}{9} c^4 d x^9 \]

[In]

Int[(d + e*x)*(a + c*x^2)^4,x]

[Out]

a^4*d*x + (4*a^3*c*d*x^3)/3 + (6*a^2*c^2*d*x^5)/5 + (4*a*c^3*d*x^7)/7 + (c^4*d*x^9)/9 + (e*(a + c*x^2)^5)/(10*
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 )^5}{10 c}+d \int \left (a+c x^2\right )^4 \, dx \\ & = \frac {e \left (a+c x^2\right )^5}{10 c}+d \int \left (a^4+4 a^3 c x^2+6 a^2 c^2 x^4+4 a c^3 x^6+c^4 x^8\right ) \, dx \\ & = a^4 d x+\frac {4}{3} a^3 c d x^3+\frac {6}{5} a^2 c^2 d x^5+\frac {4}{7} a c^3 d x^7+\frac {1}{9} c^4 d x^9+\frac {e \left (a+c x^2\right )^5}{10 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {4}{3} a^3 c d x^3+a^3 c e x^4+\frac {6}{5} a^2 c^2 d x^5+a^2 c^2 e x^6+\frac {4}{7} a c^3 d x^7+\frac {1}{2} a c^3 e x^8+\frac {1}{9} c^4 d x^9+\frac {1}{10} c^4 e x^{10} \]

[In]

Integrate[(d + e*x)*(a + c*x^2)^4,x]

[Out]

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

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33

method	result	size
gosper	\(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\)	\(97\)
default	\(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\)	\(97\)
norman	\(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\)	\(97\)
risch	\(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\)	\(97\)
parallelrisch	\(\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a \,c^{3} e \,x^{8}+\frac {4}{7} a \,c^{3} d \,x^{7}+a^{2} c^{2} e \,x^{6}+\frac {6}{5} a^{2} c^{2} d \,x^{5}+a^{3} c e \,x^{4}+\frac {4}{3} a^{3} c d \,x^{3}+\frac {1}{2} a^{4} e \,x^{2}+a^{4} d x\)	\(97\)

[In]

int((e*x+d)*(c*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=a^{4} d x + \frac {a^{4} e x^{2}}{2} + \frac {4 a^{3} c d x^{3}}{3} + a^{3} c e x^{4} + \frac {6 a^{2} c^{2} d x^{5}}{5} + a^{2} c^{2} e x^{6} + \frac {4 a c^{3} d x^{7}}{7} + \frac {a c^{3} e x^{8}}{2} + \frac {c^{4} d x^{9}}{9} + \frac {c^{4} e x^{10}}{10} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} e x^{6} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c e x^{4} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (a+c x^2\right )^4 \, dx=\frac {e\,a^4\,x^2}{2}+d\,a^4\,x+e\,a^3\,c\,x^4+\frac {4\,d\,a^3\,c\,x^3}{3}+e\,a^2\,c^2\,x^6+\frac {6\,d\,a^2\,c^2\,x^5}{5}+\frac {e\,a\,c^3\,x^8}{2}+\frac {4\,d\,a\,c^3\,x^7}{7}+\frac {e\,c^4\,x^{10}}{10}+\frac {d\,c^4\,x^9}{9} \]

[In]

int((a + c*x^2)^4*(d + e*x),x)

[Out]

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