### 3.497 $$\int (c+d x) (a+b x^2)^4 \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0306908, 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} $\frac{6}{5} a^2 b^2 c x^5+\frac{4}{3} a^3 b c x^3+a^4 c x+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9$

Antiderivative was successfully veriﬁed.

[In]

Int[(c + d*x)*(a + b*x^2)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0036308, size = 110, normalized size = 1.51 $\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{1}{9} b^4 c x^9+\frac{1}{10} b^4 d x^{10}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c + d*x)*(a + b*x^2)^4,x]

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06903, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62656, size = 220, normalized size = 3.01 \begin{align*} \frac{1}{10} x^{10} d b^{4} + \frac{1}{9} x^{9} c b^{4} + \frac{1}{2} x^{8} d b^{3} a + \frac{4}{7} x^{7} c b^{3} a + x^{6} d b^{2} a^{2} + \frac{6}{5} x^{5} c b^{2} a^{2} + x^{4} d b a^{3} + \frac{4}{3} x^{3} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082166, size = 112, normalized size = 1.53 \begin{align*} a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{b^{4} c x^{9}}{9} + \frac{b^{4} d x^{10}}{10} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27079, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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