### 3.340 $$\int \sqrt{d+e x} (b x+c x^2) \, dx$$

Optimal. Leaf size=68 $-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}$

[Out]

(2*d*(c*d - b*e)*(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 = 0.0261583, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {698} $-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(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 (b x+c x^2\right ) \, dx &=\int \left (\frac{d (c d-b e) \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 d (c d-b e) (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 = 0.0306111, size = 50, normalized size = 0.74 $\frac{2 (d+e x)^{3/2} \left (7 b e (3 e x-2 d)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(7*b*e*(-2*d + 3*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 = 0.045, size = 47, normalized size = 0.7 \begin{align*} -{\frac{-30\,c{e}^{2}{x}^{2}-42\,b{e}^{2}x+24\,cdex+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((e*x+d)^(1/2)*(c*x^2+b*x),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+14*b*d*e-8*c*d^2)/e^3

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Maxima [A]  time = 1.05738, size = 73, 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\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((e*x+d)^(1/2)*(c*x^2+b*x),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)*(e*x + d)^(3/2))/e^3

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Fricas [A]  time = 1.87582, size = 161, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 3 \,{\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} -{\left (4 \, c d^{2} e - 7 \, b d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.14416, size = 66, normalized size = 0.97 \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 (- 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((e*x+d)**(1/2)*(c*x**2+b*x),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)*(-b*d*e + c*d**2)/
(3*e**2))/e

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Giac [A]  time = 2.29449, size = 96, normalized size = 1.41 \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 )}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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