### 3.2270 $$\int \sqrt{d+e x} (a+b x+c x^2) \, dx$$

Optimal. Leaf size=75 $\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}$

[Out]

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

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Rubi [A]  time = 0.0296745, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0477906, size = 55, normalized size = 0.73 $\frac{2 (d+e x)^{3/2} \left (7 e (5 a e-2 b d+3 b e x)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(7*e*(-2*b*d + 5*a*e + 3*b*e*x) + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)

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Maple [A]  time = 0.041, size = 53, normalized size = 0.7 \begin{align*}{\frac{30\,c{e}^{2}{x}^{2}+42\,b{e}^{2}x-24\,cdex+70\,a{e}^{2}-28\,bde+16\,c{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(e*x+d)^(3/2)*(15*c*e^2*x^2+21*b*e^2*x-12*c*d*e*x+35*a*e^2-14*b*d*e+8*c*d^2)/e^3

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Maxima [A]  time = 0.97133, size = 80, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 21 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(e*x + d)^(7/2)*c - 21*(2*c*d - b*e)*(e*x + d)^(5/2) + 35*(c*d^2 - b*d*e + a*e^2)*(e*x + d)^(3/2))/e
^3

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Fricas [A]  time = 2.24467, size = 193, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 35 \, a d e^{2} + 3 \,{\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} -{\left (4 \, c d^{2} e - 7 \, b d e^{2} - 35 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*c*e^3*x^3 + 8*c*d^3 - 14*b*d^2*e + 35*a*d*e^2 + 3*(c*d*e^2 + 7*b*e^3)*x^2 - (4*c*d^2*e - 7*b*d*e^2 -
35*a*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 2.27761, size = 71, normalized size = 0.95 \begin{align*} \frac{2 \left (\frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(b*e - 2*c*d)/(5*e**2) + (d + e*x)**(3/2)*(a*e**2 - b*d*e +
c*d**2)/(3*e**2))/e

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Giac [A]  time = 1.09711, size = 111, normalized size = 1.48 \begin{align*} \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} c e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*e^(-1) + (15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*
(x*e + d)^(3/2)*d^2)*c*e^(-2) + 35*(x*e + d)^(3/2)*a)*e^(-1)