3.1263 \(\int \sqrt{b d+2 c d x} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(12*c^2*d) + (b*d + 2*c*d*x)^(7/2)/(28*c^2*d^3)

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Rubi [A]  time = 0.0221096, 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} \[ \frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(12*c^2*d) + (b*d + 2*c*d*x)^(7/2)/(28*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 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) \sqrt{b d+2 c d x}}{4 c}+\frac{(b d+2 c d x)^{5/2}}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d}+\frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0235017, size = 45, normalized size = 0.82 \[ \frac{\left (c \left (7 a+3 c x^2\right )-b^2+3 b c x\right ) (d (b+2 c x))^{3/2}}{21 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(b + 2*c*x))^(3/2)*(-b^2 + 3*b*c*x + c*(7*a + 3*c*x^2)))/(21*c^2*d)

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Maple [A]  time = 0.042, size = 46, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cx+b \right ) \left ( 3\,{c}^{2}{x}^{2}+3\,bcx+7\,ac-{b}^{2} \right ) }{21\,{c}^{2}}\sqrt{2\,cdx+bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(2*c*x+b)*(3*c^2*x^2+3*b*c*x+7*a*c-b^2)*(2*c*d*x+b*d)^(1/2)/c^2

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Maxima [A]  time = 1.02077, size = 62, normalized size = 1.13 \begin{align*} -\frac{7 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{84 \, c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(7*(2*c*d*x + b*d)^(3/2)*(b^2 - 4*a*c)*d^2 - 3*(2*c*d*x + b*d)^(7/2))/(c^2*d^3)

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Fricas [A]  time = 1.90143, size = 128, normalized size = 2.33 \begin{align*} \frac{{\left (6 \, c^{3} x^{3} + 9 \, b c^{2} x^{2} - b^{3} + 7 \, a b c +{\left (b^{2} c + 14 \, a c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{21 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(6*c^3*x^3 + 9*b*c^2*x^2 - b^3 + 7*a*b*c + (b^2*c + 14*a*c^2)*x)*sqrt(2*c*d*x + b*d)/c^2

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Sympy [A]  time = 3.23593, size = 48, normalized size = 0.87 \begin{align*} \frac{\frac{\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{12 c} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{28 c d^{2}}}{c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a),x)

[Out]

((4*a*c - b**2)*(b*d + 2*c*d*x)**(3/2)/(12*c) + (b*d + 2*c*d*x)**(7/2)/(28*c*d**2))/(c*d)

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Giac [B]  time = 1.14305, size = 157, normalized size = 2.85 \begin{align*} \frac{140 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a - \frac{14 \,{\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}{c d} + \frac{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}}}{c d^{2}}}{420 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(140*(2*c*d*x + b*d)^(3/2)*a - 14*(5*(2*c*d*x + b*d)^(3/2)*b*d - 3*(2*c*d*x + b*d)^(5/2))*b/(c*d) + (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))/(c*d^2))/(c*d)