3.337 \(\int (d+e x)^{7/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac{2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{13/2}}{13 e^3} \]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^3) + (2*c*(d + e*x)^(13/2
))/(13*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0292805, 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} \[ -\frac{2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac{2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^3) + (2*c*(d + e*x)^(13/2
))/(13*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)^{7/2} \left (b x+c x^2\right ) \, dx &=\int \left (\frac{d (c d-b e) (d+e x)^{7/2}}{e^2}+\frac{(-2 c d+b e) (d+e x)^{9/2}}{e^2}+\frac{c (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac{2 d (c d-b e) (d+e x)^{9/2}}{9 e^3}-\frac{2 (2 c d-b e) (d+e x)^{11/2}}{11 e^3}+\frac{2 c (d+e x)^{13/2}}{13 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0443229, size = 50, normalized size = 0.74 \[ \frac{2 (d+e x)^{9/2} \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(13*b*e*(-2*d + 9*e*x) + c*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 47, normalized size = 0.7 \begin{align*} -{\frac{-198\,c{e}^{2}{x}^{2}-234\,b{e}^{2}x+72\,cdex+52\,bde-16\,c{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(c*x^2+b*x),x)

[Out]

-2/1287*(e*x+d)^(9/2)*(-99*c*e^2*x^2-117*b*e^2*x+36*c*d*e*x+26*b*d*e-8*c*d^2)/e^3

________________________________________________________________________________________

Maxima [A]  time = 1.05696, size = 73, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (99 \,{\left (e x + d\right )}^{\frac{13}{2}} c - 117 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 143 \,{\left (c d^{2} - b d e\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{1287 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*c - 117*(2*c*d - b*e)*(e*x + d)^(11/2) + 143*(c*d^2 - b*d*e)*(e*x + d)^(9/2))/e^3

________________________________________________________________________________________

Fricas [B]  time = 1.88319, size = 325, normalized size = 4.78 \begin{align*} \frac{2 \,{\left (99 \, c e^{6} x^{6} + 8 \, c d^{6} - 26 \, b d^{5} e + 9 \,{\left (40 \, c d e^{5} + 13 \, b e^{6}\right )} x^{5} + 2 \,{\left (229 \, c d^{2} e^{4} + 221 \, b d e^{5}\right )} x^{4} + 2 \,{\left (106 \, c d^{3} e^{3} + 299 \, b d^{2} e^{4}\right )} x^{3} + 3 \,{\left (c d^{4} e^{2} + 104 \, b d^{3} e^{3}\right )} x^{2} -{\left (4 \, c d^{5} e - 13 \, b d^{4} e^{2}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/1287*(99*c*e^6*x^6 + 8*c*d^6 - 26*b*d^5*e + 9*(40*c*d*e^5 + 13*b*e^6)*x^5 + 2*(229*c*d^2*e^4 + 221*b*d*e^5)*
x^4 + 2*(106*c*d^3*e^3 + 299*b*d^2*e^4)*x^3 + 3*(c*d^4*e^2 + 104*b*d^3*e^3)*x^2 - (4*c*d^5*e - 13*b*d^4*e^2)*x
)*sqrt(e*x + d)/e^3

________________________________________________________________________________________

Sympy [A]  time = 15.1002, size = 292, normalized size = 4.29 \begin{align*} \begin{cases} - \frac{4 b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 c d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 c d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 c d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 c d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 c d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 c d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 c e^{3} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\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)**(7/2)*(c*x**2+b*x),x)

[Out]

Piecewise((-4*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*b*d**4*x*sqrt(d + e*x)/(99*e) + 16*b*d**3*x**2*sqrt(d + e*x)/
33 + 92*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*b*e**3*x**5*sqrt(d + e*x)/11 +
16*c*d**6*sqrt(d + e*x)/(1287*e**3) - 8*c*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*c*d**4*x**2*sqrt(d + e*x)/(429*
e) + 424*c*d**3*x**3*sqrt(d + e*x)/1287 + 916*c*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*c*d*e**2*x**5*sqrt(d + e*x
)/143 + 2*c*e**3*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(b*x**2/2 + c*x**3/3), True))

________________________________________________________________________________________

Giac [B]  time = 1.46589, size = 598, normalized size = 8.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*d^3*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^
(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*c*d^3*e^(-2) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d
)^(3/2)*d^2)*b*d^2*e^(-1) + 429*(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^2*e^(-2) + 429*(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)*b*d*e^(-1) + 39*(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*d*e^(-2) + 13*(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)*b*e^(-1) + 5*(6
93*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(
x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*c*e^(-2))*e^(-1)