### 3.1206 $$\int (b d+2 c d x) (a+b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=19 $\frac{2}{5} d \left (a+b x+c x^2\right )^{5/2}$

[Out]

(2*d*(a + b*x + c*x^2)^(5/2))/5

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Rubi [A]  time = 0.0066193, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {629} $\frac{2}{5} d \left (a+b x+c x^2\right )^{5/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(2*d*(a + b*x + c*x^2)^(5/2))/5

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{5} d \left (a+b x+c x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0144772, size = 18, normalized size = 0.95 $\frac{2}{5} d (a+x (b+c x))^{5/2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(2*d*(a + x*(b + c*x))^(5/2))/5

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Maple [A]  time = 0.042, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,d}{5} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)*(c*x^2+b*x+a)^(3/2),x)

[Out]

2/5*d*(c*x^2+b*x+a)^(5/2)

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Maxima [A]  time = 1.12318, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{5} \,{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}} d \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x^2 + b*x + a)^(5/2)*d

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Fricas [B]  time = 2.63561, size = 128, normalized size = 6.74 \begin{align*} \frac{2}{5} \,{\left (c^{2} d x^{4} + 2 \, b c d x^{3} + 2 \, a b d x +{\left (b^{2} + 2 \, a c\right )} d x^{2} + a^{2} d\right )} \sqrt{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c^2*d*x^4 + 2*b*c*d*x^3 + 2*a*b*d*x + (b^2 + 2*a*c)*d*x^2 + a^2*d)*sqrt(c*x^2 + b*x + a)

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Sympy [B]  time = 0.939766, size = 146, normalized size = 7.68 \begin{align*} \frac{2 a^{2} d \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a b d x \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a c d x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 b^{2} d x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{4 b c d x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 c^{2} d x^{4} \sqrt{a + b x + c x^{2}}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*d*sqrt(a + b*x + c*x**2)/5 + 4*a*b*d*x*sqrt(a + b*x + c*x**2)/5 + 4*a*c*d*x**2*sqrt(a + b*x + c*x**2)/5
+ 2*b**2*d*x**2*sqrt(a + b*x + c*x**2)/5 + 4*b*c*d*x**3*sqrt(a + b*x + c*x**2)/5 + 2*c**2*d*x**4*sqrt(a + b*x
+ c*x**2)/5

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Giac [B]  time = 1.22153, size = 88, normalized size = 4.63 \begin{align*} \frac{2}{5} \,{\left (a^{2} d +{\left (2 \, a b d +{\left ({\left (c^{2} d x + 2 \, b c d\right )} x + \frac{b^{2} c^{4} d + 2 \, a c^{5} d}{c^{4}}\right )} x\right )} x\right )} \sqrt{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a^2*d + (2*a*b*d + ((c^2*d*x + 2*b*c*d)*x + (b^2*c^4*d + 2*a*c^5*d)/c^4)*x)*x)*sqrt(c*x^2 + b*x + a)