3.1204 \(\int (b d+2 c d x)^3 (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=59 \[ \frac{4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(5/2))/35 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(5/2))/7

________________________________________________________________________________________

Rubi [A]  time = 0.0269461, 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} \[ \frac{4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(5/2))/35 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(5/2))/7

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 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{1}{7} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^{3/2} \, dx\\ &=\frac{4}{35} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0466772, size = 44, normalized size = 0.75 \[ \frac{2}{35} d^3 (a+x (b+c x))^{5/2} \left (4 c \left (5 c x^2-2 a\right )+7 b^2+20 b c x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^3*(a + x*(b + c*x))^(5/2)*(7*b^2 + 20*b*c*x + 4*c*(-2*a + 5*c*x^2)))/35

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -40\,{c}^{2}{x}^{2}-40\,bcx+16\,ac-14\,{b}^{2} \right ){d}^{3}}{35} \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)^3*(c*x^2+b*x+a)^(3/2),x)

[Out]

-2/35*(c*x^2+b*x+a)^(5/2)*(-20*c^2*x^2-20*b*c*x+8*a*c-7*b^2)*d^3

________________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.66994, size = 320, normalized size = 5.42 \begin{align*} \frac{2}{35} \,{\left (20 \, c^{4} d^{3} x^{6} + 60 \, b c^{3} d^{3} x^{5} +{\left (67 \, b^{2} c^{2} + 32 \, a c^{3}\right )} d^{3} x^{4} + 2 \,{\left (17 \, b^{3} c + 32 \, a b c^{2}\right )} d^{3} x^{3} +{\left (7 \, b^{4} + 46 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{3} x^{2} + 2 \,{\left (7 \, a b^{3} + 2 \, a^{2} b c\right )} d^{3} x +{\left (7 \, a^{2} b^{2} - 8 \, a^{3} 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)^(3/2),x, algorithm="fricas")

[Out]

2/35*(20*c^4*d^3*x^6 + 60*b*c^3*d^3*x^5 + (67*b^2*c^2 + 32*a*c^3)*d^3*x^4 + 2*(17*b^3*c + 32*a*b*c^2)*d^3*x^3
+ (7*b^4 + 46*a*b^2*c + 4*a^2*c^2)*d^3*x^2 + 2*(7*a*b^3 + 2*a^2*b*c)*d^3*x + (7*a^2*b^2 - 8*a^3*c)*d^3)*sqrt(c
*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [B]  time = 1.83522, size = 371, normalized size = 6.29 \begin{align*} - \frac{16 a^{3} c d^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{2 a^{2} b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{8 a^{2} b c d^{3} x \sqrt{a + b x + c x^{2}}}{35} + \frac{8 a^{2} c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{35} + \frac{4 a b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{5} + \frac{92 a b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{35} + \frac{128 a b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{64 a c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{35} + \frac{2 b^{4} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{68 b^{3} c d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{134 b^{2} c^{2} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{35} + \frac{24 b c^{3} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{8 c^{4} d^{3} 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)**3*(c*x**2+b*x+a)**(3/2),x)

[Out]

-16*a**3*c*d**3*sqrt(a + b*x + c*x**2)/35 + 2*a**2*b**2*d**3*sqrt(a + b*x + c*x**2)/5 + 8*a**2*b*c*d**3*x*sqrt
(a + b*x + c*x**2)/35 + 8*a**2*c**2*d**3*x**2*sqrt(a + b*x + c*x**2)/35 + 4*a*b**3*d**3*x*sqrt(a + b*x + c*x**
2)/5 + 92*a*b**2*c*d**3*x**2*sqrt(a + b*x + c*x**2)/35 + 128*a*b*c**2*d**3*x**3*sqrt(a + b*x + c*x**2)/35 + 64
*a*c**3*d**3*x**4*sqrt(a + b*x + c*x**2)/35 + 2*b**4*d**3*x**2*sqrt(a + b*x + c*x**2)/5 + 68*b**3*c*d**3*x**3*
sqrt(a + b*x + c*x**2)/35 + 134*b**2*c**2*d**3*x**4*sqrt(a + b*x + c*x**2)/35 + 24*b*c**3*d**3*x**5*sqrt(a + b
*x + c*x**2)/7 + 8*c**4*d**3*x**6*sqrt(a + b*x + c*x**2)/7

________________________________________________________________________________________

Giac [B]  time = 1.13857, size = 269, normalized size = 4.56 \begin{align*} \frac{2}{35} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left (20 \,{\left (c^{4} d^{3} x + 3 \, b c^{3} d^{3}\right )} x + \frac{67 \, b^{2} c^{8} d^{3} + 32 \, a c^{9} d^{3}}{c^{6}}\right )} x + \frac{2 \,{\left (17 \, b^{3} c^{7} d^{3} + 32 \, a b c^{8} d^{3}\right )}}{c^{6}}\right )} x + \frac{7 \, b^{4} c^{6} d^{3} + 46 \, a b^{2} c^{7} d^{3} + 4 \, a^{2} c^{8} d^{3}}{c^{6}}\right )} x + \frac{2 \,{\left (7 \, a b^{3} c^{6} d^{3} + 2 \, a^{2} b c^{7} d^{3}\right )}}{c^{6}}\right )} x + \frac{7 \, a^{2} b^{2} c^{6} d^{3} - 8 \, a^{3} c^{7} d^{3}}{c^{6}}\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)^(3/2),x, algorithm="giac")

[Out]

2/35*sqrt(c*x^2 + b*x + a)*(((((20*(c^4*d^3*x + 3*b*c^3*d^3)*x + (67*b^2*c^8*d^3 + 32*a*c^9*d^3)/c^6)*x + 2*(1
7*b^3*c^7*d^3 + 32*a*b*c^8*d^3)/c^6)*x + (7*b^4*c^6*d^3 + 46*a*b^2*c^7*d^3 + 4*a^2*c^8*d^3)/c^6)*x + 2*(7*a*b^
3*c^6*d^3 + 2*a^2*b*c^7*d^3)/c^6)*x + (7*a^2*b^2*c^6*d^3 - 8*a^3*c^7*d^3)/c^6)