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

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)

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Rubi [A]  time = 0.101854, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac{2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{13/2}}{13 e^3} \]

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)

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Rubi in Sympy [A]  time = 13.6975, size = 63, normalized size = 0.93 \[ \frac{2 c \left (d + e x\right )^{\frac{13}{2}}}{13 e^{3}} - \frac{2 d \left (d + e x\right )^{\frac{9}{2}} \left (b e - c d\right )}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )}{11 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)*(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.087173, 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)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.7 \[ -{\frac{-198\,c{e}^{2}{x}^{2}-234\,b{e}^{2}x+72\,cdex+52\,bde-16\,c{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

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

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Maxima [A]  time = 0.688245, size = 73, normalized size = 1.07 \[ \frac{2 \,{\left (99 \,{\left (e x + d\right )}^{\frac{13}{2}} c - 117 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 143 \,{\left (c d^{2} - b d e\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.216216, size = 193, normalized size = 2.84 \[ \frac{2 \,{\left (99 \, c e^{6} x^{6} + 8 \, c d^{6} - 26 \, b d^{5} e + 9 \,{\left (40 \, c d e^{5} + 13 \, b e^{6}\right )} x^{5} + 2 \,{\left (229 \, c d^{2} e^{4} + 221 \, b d e^{5}\right )} x^{4} + 2 \,{\left (106 \, c d^{3} e^{3} + 299 \, b d^{2} e^{4}\right )} x^{3} + 3 \,{\left (c d^{4} e^{2} + 104 \, b d^{3} e^{3}\right )} x^{2} -{\left (4 \, c d^{5} e - 13 \, b d^{4} e^{2}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 23.3718, size = 292, normalized size = 4.29 \[ \begin{cases} - \frac{4 b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 c d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 c d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 c d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 c d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 c d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 c d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 c e^{3} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\frac{b x^{2}}{2} + \frac{c x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]

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))

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GIAC/XCAS [A]  time = 0.214441, size = 679, normalized size = 9.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^(7/2),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)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*c
*d^3*e^(-14) + 1287*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x
*e + d)^(3/2)*d^2*e^12)*b*d^2*e^(-13) + 429*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e
+ d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)
*c*d^2*e^(-26) + 429*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189
*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*b*d*e^(-25) + 39*(315*
(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e
^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*c*d*e^(-42)
+ 13*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^
(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*
b*e^(-41) + 5*(693*(x*e + d)^(13/2)*e^60 - 4095*(x*e + d)^(11/2)*d*e^60 + 10010*
(x*e + d)^(9/2)*d^2*e^60 - 12870*(x*e + d)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(5/2)
*d^4*e^60 - 3003*(x*e + d)^(3/2)*d^5*e^60)*c*e^(-62))*e^(-1)