### 3.1221 $$\int (b d+2 c d x) (a+b x+c x^2)^{5/2} \, dx$$

Optimal. Leaf size=19 $\frac{2}{7} d \left (a+b x+c x^2\right )^{7/2}$

[Out]

(2*d*(a + b*x + c*x^2)^(7/2))/7

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Rubi [A]  time = 0.0066538, 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} $\frac{2}{7} d \left (a+b x+c x^2\right )^{7/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*d*(a + b*x + c*x^2)^(7/2))/7

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 )^{5/2} \, dx &=\frac{2}{7} d \left (a+b x+c x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0127724, size = 18, normalized size = 0.95 $\frac{2}{7} d (a+x (b+c x))^{7/2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*d*(a + x*(b + c*x))^(7/2))/7

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Maple [A]  time = 0.041, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,d}{7} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{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)^(5/2),x)

[Out]

2/7*d*(c*x^2+b*x+a)^(7/2)

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Maxima [A]  time = 1.09422, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{7} \,{\left (c x^{2} + b x + a\right )}^{\frac{7}{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)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c*x^2 + b*x + a)^(7/2)*d

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Fricas [B]  time = 2.8411, size = 207, normalized size = 10.89 \begin{align*} \frac{2}{7} \,{\left (c^{3} d x^{6} + 3 \, b c^{2} d x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} d x^{4} + 3 \, a^{2} b d x +{\left (b^{3} + 6 \, a b c\right )} d x^{3} + a^{3} d + 3 \,{\left (a b^{2} + a^{2} c\right )} d x^{2}\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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.38953, size = 260, normalized size = 13.68 \begin{align*} \frac{2 a^{3} d \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} b d x \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} c d x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a b^{2} d x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{12 a b c d x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a c^{2} d x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 b^{3} d x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b^{2} c d x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b c^{2} d x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 c^{3} d x^{6} \sqrt{a + b x + c x^{2}}}{7} \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)**(5/2),x)

[Out]

2*a**3*d*sqrt(a + b*x + c*x**2)/7 + 6*a**2*b*d*x*sqrt(a + b*x + c*x**2)/7 + 6*a**2*c*d*x**2*sqrt(a + b*x + c*x
**2)/7 + 6*a*b**2*d*x**2*sqrt(a + b*x + c*x**2)/7 + 12*a*b*c*d*x**3*sqrt(a + b*x + c*x**2)/7 + 6*a*c**2*d*x**4
*sqrt(a + b*x + c*x**2)/7 + 2*b**3*d*x**3*sqrt(a + b*x + c*x**2)/7 + 6*b**2*c*d*x**4*sqrt(a + b*x + c*x**2)/7
+ 6*b*c**2*d*x**5*sqrt(a + b*x + c*x**2)/7 + 2*c**3*d*x**6*sqrt(a + b*x + c*x**2)/7

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Giac [B]  time = 1.19344, size = 161, normalized size = 8.47 \begin{align*} \frac{2}{7} \,{\left (a^{3} d +{\left (3 \, a^{2} b d +{\left ({\left ({\left ({\left (c^{3} d x + 3 \, b c^{2} d\right )} x + \frac{3 \,{\left (b^{2} c^{7} d + a c^{8} d\right )}}{c^{6}}\right )} x + \frac{b^{3} c^{6} d + 6 \, a b c^{7} d}{c^{6}}\right )} x + \frac{3 \,{\left (a b^{2} c^{6} d + a^{2} c^{7} d\right )}}{c^{6}}\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)^(5/2),x, algorithm="giac")

[Out]

2/7*(a^3*d + (3*a^2*b*d + ((((c^3*d*x + 3*b*c^2*d)*x + 3*(b^2*c^7*d + a*c^8*d)/c^6)*x + (b^3*c^6*d + 6*a*b*c^7
*d)/c^6)*x + 3*(a*b^2*c^6*d + a^2*c^7*d)/c^6)*x)*x)*sqrt(c*x^2 + b*x + a)