3.1192 $$\int (b d+2 c d x)^3 \sqrt{a+b x+c x^2} \, dx$$

Optimal. Leaf size=59 $\frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}$

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(3/2))/15 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(3/2))/5

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Rubi [A]  time = 0.0262283, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {692, 629} $\frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(3/2))/15 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(3/2))/5

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

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)^3 \sqrt{a+b x+c x^2} \, dx &=\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{1}{5} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \sqrt{a+b x+c x^2} \, dx\\ &=\frac{4}{15} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0378078, size = 44, normalized size = 0.75 $\frac{2}{15} d^3 (a+x (b+c x))^{3/2} \left (4 c \left (3 c x^2-2 a\right )+5 b^2+12 b c x\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*d^3*(a + x*(b + c*x))^(3/2)*(5*b^2 + 12*b*c*x + 4*c*(-2*a + 3*c*x^2)))/15

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Maple [A]  time = 0.045, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -24\,{c}^{2}{x}^{2}-24\,bcx+16\,ac-10\,{b}^{2} \right ){d}^{3}}{15} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^(1/2),x)

[Out]

-2/15*(c*x^2+b*x+a)^(3/2)*(-12*c^2*x^2-12*b*c*x+8*a*c-5*b^2)*d^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.25002, size = 198, normalized size = 3.36 \begin{align*} \frac{2}{15} \,{\left (12 \, c^{3} d^{3} x^{4} + 24 \, b c^{2} d^{3} x^{3} +{\left (17 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} x^{2} +{\left (5 \, b^{3} + 4 \, a b c\right )} d^{3} x +{\left (5 \, a b^{2} - 8 \, a^{2} c\right )} d^{3}\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)^3*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/15*(12*c^3*d^3*x^4 + 24*b*c^2*d^3*x^3 + (17*b^2*c + 4*a*c^2)*d^3*x^2 + (5*b^3 + 4*a*b*c)*d^3*x + (5*a*b^2 -
8*a^2*c)*d^3)*sqrt(c*x^2 + b*x + a)

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Sympy [B]  time = 0.497091, size = 216, normalized size = 3.66 \begin{align*} - \frac{16 a^{2} c d^{3} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 a b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{3} + \frac{8 a b c d^{3} x \sqrt{a + b x + c x^{2}}}{15} + \frac{8 a c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{34 b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{16 b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{8 c^{3} d^{3} 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)**3*(c*x**2+b*x+a)**(1/2),x)

[Out]

-16*a**2*c*d**3*sqrt(a + b*x + c*x**2)/15 + 2*a*b**2*d**3*sqrt(a + b*x + c*x**2)/3 + 8*a*b*c*d**3*x*sqrt(a + b
*x + c*x**2)/15 + 8*a*c**2*d**3*x**2*sqrt(a + b*x + c*x**2)/15 + 2*b**3*d**3*x*sqrt(a + b*x + c*x**2)/3 + 34*b
**2*c*d**3*x**2*sqrt(a + b*x + c*x**2)/15 + 16*b*c**2*d**3*x**3*sqrt(a + b*x + c*x**2)/5 + 8*c**3*d**3*x**4*sq
rt(a + b*x + c*x**2)/5

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Giac [B]  time = 1.19079, size = 163, normalized size = 2.76 \begin{align*} \frac{2}{15} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left (12 \,{\left (c^{3} d^{3} x + 2 \, b c^{2} d^{3}\right )} x + \frac{17 \, b^{2} c^{5} d^{3} + 4 \, a c^{6} d^{3}}{c^{4}}\right )} x + \frac{5 \, b^{3} c^{4} d^{3} + 4 \, a b c^{5} d^{3}}{c^{4}}\right )} x + \frac{5 \, a b^{2} c^{4} d^{3} - 8 \, a^{2} c^{5} d^{3}}{c^{4}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/15*sqrt(c*x^2 + b*x + a)*(((12*(c^3*d^3*x + 2*b*c^2*d^3)*x + (17*b^2*c^5*d^3 + 4*a*c^6*d^3)/c^4)*x + (5*b^3*
c^4*d^3 + 4*a*b*c^5*d^3)/c^4)*x + (5*a*b^2*c^4*d^3 - 8*a^2*c^5*d^3)/c^4)