3.2268 \(\int (d+e x)^{5/2} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=75 \[ \frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}{7 e^3}-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) + (2*c*(d + e*
x)^(11/2))/(11*e^3)

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Rubi [A]  time = 0.0381153, 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} \[ \frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}{7 e^3}-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + b*x + c*x^2),x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) + (2*c*(d + e*
x)^(11/2))/(11*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 (d+e x)^{5/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{e^2}+\frac{(-2 c d+b e) (d+e x)^{7/2}}{e^2}+\frac{c (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac{2 (2 c d-b e) (d+e x)^{9/2}}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0620969, size = 55, normalized size = 0.73 \[ \frac{2 (d+e x)^{7/2} \left (11 e (9 a e-2 b d+7 b e x)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(11*e*(-2*b*d + 9*a*e + 7*b*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)

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Maple [A]  time = 0.042, size = 53, normalized size = 0.7 \begin{align*}{\frac{126\,c{e}^{2}{x}^{2}+154\,b{e}^{2}x-56\,cdex+198\,a{e}^{2}-44\,bde+16\,c{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+b*x+a),x)

[Out]

2/693*(e*x+d)^(7/2)*(63*c*e^2*x^2+77*b*e^2*x-28*c*d*e*x+99*a*e^2-22*b*d*e+8*c*d^2)/e^3

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Maxima [A]  time = 1.04428, size = 80, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} c - 77 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*c - 77*(2*c*d - b*e)*(e*x + d)^(9/2) + 99*(c*d^2 - b*d*e + a*e^2)*(e*x + d)^(7/2))/
e^3

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Fricas [B]  time = 2.1901, size = 340, normalized size = 4.53 \begin{align*} \frac{2 \,{\left (63 \, c e^{5} x^{5} + 8 \, c d^{5} - 22 \, b d^{4} e + 99 \, a d^{3} e^{2} + 7 \,{\left (23 \, c d e^{4} + 11 \, b e^{5}\right )} x^{4} +{\left (113 \, c d^{2} e^{3} + 209 \, b d e^{4} + 99 \, a e^{5}\right )} x^{3} + 3 \,{\left (c d^{3} e^{2} + 55 \, b d^{2} e^{3} + 99 \, a d e^{4}\right )} x^{2} -{\left (4 \, c d^{4} e - 11 \, b d^{3} e^{2} - 297 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

2/693*(63*c*e^5*x^5 + 8*c*d^5 - 22*b*d^4*e + 99*a*d^3*e^2 + 7*(23*c*d*e^4 + 11*b*e^5)*x^4 + (113*c*d^2*e^3 + 2
09*b*d*e^4 + 99*a*e^5)*x^3 + 3*(c*d^3*e^2 + 55*b*d^2*e^3 + 99*a*d*e^4)*x^2 - (4*c*d^4*e - 11*b*d^3*e^2 - 297*a
*d^2*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.35191, size = 326, normalized size = 4.35 \begin{align*} \begin{cases} \frac{2 a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a d^{2} x \sqrt{d + e x}}{7} + \frac{6 a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 b e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 c d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 c d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 c d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 c d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 c d e x^{4} \sqrt{d + e x}}{99} + \frac{2 c e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(c*x**2+b*x+a),x)

[Out]

Piecewise((2*a*d**3*sqrt(d + e*x)/(7*e) + 6*a*d**2*x*sqrt(d + e*x)/7 + 6*a*d*e*x**2*sqrt(d + e*x)/7 + 2*a*e**2
*x**3*sqrt(d + e*x)/7 - 4*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*b*d**3*x*sqrt(d + e*x)/(63*e) + 10*b*d**2*x**2*sq
rt(d + e*x)/21 + 38*b*d*e*x**3*sqrt(d + e*x)/63 + 2*b*e**2*x**4*sqrt(d + e*x)/9 + 16*c*d**5*sqrt(d + e*x)/(693
*e**3) - 8*c*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*c*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*c*d**2*x**3*sqrt(d +
e*x)/693 + 46*c*d*e*x**4*sqrt(d + e*x)/99 + 2*c*e**2*x**5*sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(a*x + b*x**2
/2 + c*x**3/3), True))

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Giac [B]  time = 1.15002, size = 498, normalized size = 6.64 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b d^{2} e^{\left (-1\right )} + 33 \,{\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 d^{2} e^{\left (-2\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2} + 66 \,{\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 )} b d e^{\left (-1\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} c d e^{\left (-2\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b e^{\left (-1\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} c e^{\left (-2\right )} + 33 \,{\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 )} a\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*d^2*e^(-1) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/
2)*d + 35*(x*e + d)^(3/2)*d^2)*c*d^2*e^(-2) + 1155*(x*e + d)^(3/2)*a*d^2 + 66*(15*(x*e + d)^(7/2) - 42*(x*e +
d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b*d*e^(-1) + 22*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e +
d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*c*d*e^(-2) + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*d + 11*(3
5*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b*e^(-1) + (315
*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e +
d)^(3/2)*d^4)*c*e^(-2) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a)*e^(-1)