3.275 \(\int (A+B x) (a+c x^2)^4 \, dx\)

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

[Out]

a^4*A*x + (4*a^3*A*c*x^3)/3 + (6*a^2*A*c^2*x^5)/5 + (4*a*A*c^3*x^7)/7 + (A*c^4*x^9)/9 + (B*(a + c*x^2)^5)/(10*
c)

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Rubi [A]  time = 0.0250958, 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} \[ \frac{6}{5} a^2 A c^2 x^5+\frac{4}{3} a^3 A c x^3+a^4 A x+\frac{4}{7} a A c^3 x^7+\frac{B \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} A c^4 x^9 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

Mathematica [A]  time = 0.0029984, size = 110, normalized size = 1.51 \[ \frac{6}{5} a^2 A c^2 x^5+\frac{4}{3} a^3 A c x^3+a^4 A x+a^2 B c^2 x^6+a^3 B c x^4+\frac{1}{2} a^4 B x^2+\frac{4}{7} a A c^3 x^7+\frac{1}{2} a B c^3 x^8+\frac{1}{9} A c^4 x^9+\frac{1}{10} B c^4 x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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Maple [A]  time = 0.001, size = 97, normalized size = 1.3 \begin{align*}{\frac{B{c}^{4}{x}^{10}}{10}}+{\frac{A{c}^{4}{x}^{9}}{9}}+{\frac{aB{c}^{3}{x}^{8}}{2}}+{\frac{4\,aA{c}^{3}{x}^{7}}{7}}+{a}^{2}B{c}^{2}{x}^{6}+{\frac{6\,{a}^{2}A{c}^{2}{x}^{5}}{5}}+{a}^{3}Bc{x}^{4}+{\frac{4\,{a}^{3}Ac{x}^{3}}{3}}+{\frac{{a}^{4}B{x}^{2}}{2}}+{a}^{4}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.03723, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.2478, size = 220, normalized size = 3.01 \begin{align*} \frac{1}{10} x^{10} c^{4} B + \frac{1}{9} x^{9} c^{4} A + \frac{1}{2} x^{8} c^{3} a B + \frac{4}{7} x^{7} c^{3} a A + x^{6} c^{2} a^{2} B + \frac{6}{5} x^{5} c^{2} a^{2} A + x^{4} c a^{3} B + \frac{4}{3} x^{3} c a^{3} A + \frac{1}{2} x^{2} a^{4} B + x a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.118268, size = 112, normalized size = 1.53 \begin{align*} A a^{4} x + \frac{4 A a^{3} c x^{3}}{3} + \frac{6 A a^{2} c^{2} x^{5}}{5} + \frac{4 A a c^{3} x^{7}}{7} + \frac{A c^{4} x^{9}}{9} + \frac{B a^{4} x^{2}}{2} + B a^{3} c x^{4} + B a^{2} c^{2} x^{6} + \frac{B a c^{3} x^{8}}{2} + \frac{B c^{4} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**4,x)

[Out]

A*a**4*x + 4*A*a**3*c*x**3/3 + 6*A*a**2*c**2*x**5/5 + 4*A*a*c**3*x**7/7 + A*c**4*x**9/9 + B*a**4*x**2/2 + B*a*
*3*c*x**4 + B*a**2*c**2*x**6 + B*a*c**3*x**8/2 + B*c**4*x**10/10

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Giac [A]  time = 1.12445, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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