3.1262 \(\int (b d+2 c d x)^{3/2} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{9/2}}{36 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d} \]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2))/(20*c^2*d) + (b*d + 2*c*d*x)^(9/2)/(36*c^2*d^3)

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Rubi [A]  time = 0.0227016, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ \frac{(b d+2 c d x)^{9/2}}{36 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2))/(20*c^2*d) + (b*d + 2*c*d*x)^(9/2)/(36*c^2*d^3)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{4 c}+\frac{(b d+2 c d x)^{7/2}}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d}+\frac{(b d+2 c d x)^{9/2}}{36 c^2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0292183, size = 45, normalized size = 0.82 \[ \frac{\left (c \left (9 a+5 c x^2\right )-b^2+5 b c x\right ) (d (b+2 c x))^{5/2}}{45 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(b + 2*c*x))^(5/2)*(-b^2 + 5*b*c*x + c*(9*a + 5*c*x^2)))/(45*c^2*d)

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Maple [A]  time = 0.043, size = 46, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cx+b \right ) \left ( 5\,{c}^{2}{x}^{2}+5\,bcx+9\,ac-{b}^{2} \right ) }{45\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(2*c*x+b)*(5*c^2*x^2+5*b*c*x+9*a*c-b^2)*(2*c*d*x+b*d)^(3/2)/c^2

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Maxima [A]  time = 1.11853, size = 62, normalized size = 1.13 \begin{align*} -\frac{9 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{180 \, c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/180*(9*(2*c*d*x + b*d)^(5/2)*(b^2 - 4*a*c)*d^2 - 5*(2*c*d*x + b*d)^(9/2))/(c^2*d^3)

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Fricas [A]  time = 2.0613, size = 194, normalized size = 3.53 \begin{align*} \frac{{\left (20 \, c^{4} d x^{4} + 40 \, b c^{3} d x^{3} + 3 \,{\left (7 \, b^{2} c^{2} + 12 \, a c^{3}\right )} d x^{2} +{\left (b^{3} c + 36 \, a b c^{2}\right )} d x -{\left (b^{4} - 9 \, a b^{2} c\right )} d\right )} \sqrt{2 \, c d x + b d}}{45 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(20*c^4*d*x^4 + 40*b*c^3*d*x^3 + 3*(7*b^2*c^2 + 12*a*c^3)*d*x^2 + (b^3*c + 36*a*b*c^2)*d*x - (b^4 - 9*a*b
^2*c)*d)*sqrt(2*c*d*x + b*d)/c^2

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Sympy [A]  time = 10.2782, size = 274, normalized size = 4.98 \begin{align*} a b d \left (\begin{cases} x \sqrt{b d} & \text{for}\: c = 0 \\0 & \text{for}\: d = 0 \\\frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{3 c d} & \text{otherwise} \end{cases}\right ) + \frac{a \left (- \frac{b d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{5}\right )}{c d} + \frac{b^{2} \left (- \frac{b d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{5}\right )}{2 c^{2} d} + \frac{3 b \left (\frac{b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} - \frac{2 b d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{7}\right )}{4 c^{2} d^{2}} + \frac{- \frac{b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} + \frac{3 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} - \frac{3 b d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7} + \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}}}{9}}{4 c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b*d*Piecewise((x*sqrt(b*d), Eq(c, 0)), (0, Eq(d, 0)), ((b*d + 2*c*d*x)**(3/2)/(3*c*d), True)) + a*(-b*d*(b*d
+ 2*c*d*x)**(3/2)/3 + (b*d + 2*c*d*x)**(5/2)/5)/(c*d) + b**2*(-b*d*(b*d + 2*c*d*x)**(3/2)/3 + (b*d + 2*c*d*x)
**(5/2)/5)/(2*c**2*d) + 3*b*(b**2*d**2*(b*d + 2*c*d*x)**(3/2)/3 - 2*b*d*(b*d + 2*c*d*x)**(5/2)/5 + (b*d + 2*c*
d*x)**(7/2)/7)/(4*c**2*d**2) + (-b**3*d**3*(b*d + 2*c*d*x)**(3/2)/3 + 3*b**2*d**2*(b*d + 2*c*d*x)**(5/2)/5 - 3
*b*d*(b*d + 2*c*d*x)**(7/2)/7 + (b*d + 2*c*d*x)**(9/2)/9)/(4*c**2*d**3)

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Giac [B]  time = 1.17555, size = 308, normalized size = 5.6 \begin{align*} \frac{420 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b - \frac{84 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} a}{d} - \frac{42 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} b^{2}}{c d} + \frac{9 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{2} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b d + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )} b}{c d^{2}} - \frac{105 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} d^{3} - 189 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} d^{2} + 135 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b d - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{c d^{3}}}{1260 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(420*(2*c*d*x + b*d)^(3/2)*a*b - 84*(5*(2*c*d*x + b*d)^(3/2)*b*d - 3*(2*c*d*x + b*d)^(5/2))*a/d - 42*(5
*(2*c*d*x + b*d)^(3/2)*b*d - 3*(2*c*d*x + b*d)^(5/2))*b^2/(c*d) + 9*(35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 - 42*(2*
c*d*x + b*d)^(5/2)*b*d + 15*(2*c*d*x + b*d)^(7/2))*b/(c*d^2) - (105*(2*c*d*x + b*d)^(3/2)*b^3*d^3 - 189*(2*c*d
*x + b*d)^(5/2)*b^2*d^2 + 135*(2*c*d*x + b*d)^(7/2)*b*d - 35*(2*c*d*x + b*d)^(9/2))/(c*d^3))/c