3.494 \(\int (d+e x) (a+c x^2)^4 \, dx\)

Optimal. Leaf size=73 \[ \frac{6}{5} a^2 c^2 d x^5+\frac{4}{3} a^3 c d x^3+a^4 d x+\frac{4}{7} a c^3 d x^7+\frac{e \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} c^4 d x^9 \]

[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)

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Rubi [A]  time = 0.0288876, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ \frac{6}{5} a^2 c^2 d x^5+\frac{4}{3} a^3 c d x^3+a^4 d x+\frac{4}{7} a c^3 d x^7+\frac{e \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} c^4 d x^9 \]

Antiderivative was successfully verified.

[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 641

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]

Rule 194

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]

Rubi steps

\begin{align*} \int (d+e x) \left (a+c x^2\right )^4 \, dx &=\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]  time = 0.0029015, size = 110, normalized size = 1.51 \[ \frac{6}{5} a^2 c^2 d x^5+a^2 c^2 e x^6+\frac{4}{3} a^3 c d x^3+a^3 c e x^4+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{7} a c^3 d x^7+\frac{1}{2} a c^3 e x^8+\frac{1}{9} c^4 d x^9+\frac{1}{10} c^4 e x^{10} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.042, size = 97, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}e{x}^{10}}{10}}+{\frac{{c}^{4}d{x}^{9}}{9}}+{\frac{ea{c}^{3}{x}^{8}}{2}}+{\frac{4\,a{c}^{3}d{x}^{7}}{7}}+e{a}^{2}{c}^{2}{x}^{6}+{\frac{6\,{a}^{2}{c}^{2}d{x}^{5}}{5}}+e{a}^{3}c{x}^{4}+{\frac{4\,{a}^{3}cd{x}^{3}}{3}}+{\frac{e{a}^{4}{x}^{2}}{2}}+{a}^{4}dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20309, size = 130, normalized size = 1.78 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.71082, size = 220, normalized size = 3.01 \begin{align*} \frac{1}{10} x^{10} e c^{4} + \frac{1}{9} x^{9} d c^{4} + \frac{1}{2} x^{8} e c^{3} a + \frac{4}{7} x^{7} d c^{3} a + x^{6} e c^{2} a^{2} + \frac{6}{5} x^{5} d c^{2} a^{2} + x^{4} e c a^{3} + \frac{4}{3} x^{3} d c a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.087271, size = 112, normalized size = 1.53 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.32896, size = 136, normalized size = 1.86 \begin{align*} \frac{1}{10} \, c^{4} x^{10} e + \frac{1}{9} \, c^{4} d x^{9} + \frac{1}{2} \, a c^{3} x^{8} e + \frac{4}{7} \, a c^{3} d x^{7} + a^{2} c^{2} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d x^{5} + a^{3} c x^{4} e + \frac{4}{3} \, a^{3} c d x^{3} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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