3.1610 \(\int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac{2 b^2 (d+e x)^{13/2}}{13 e^3} \]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(11/2))/(
11*e^3) + (2*b^2*(d + e*x)^(13/2))/(13*e^3)

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Rubi [A]  time = 0.0882664, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac{2 b^2 (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(11/2))/(
11*e^3) + (2*b^2*(d + e*x)^(13/2))/(13*e^3)

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Rubi in Sympy [A]  time = 30.8796, size = 65, normalized size = 0.92 \[ \frac{2 b^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{3}} + \frac{4 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )}{11 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}}{9 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(d + e*x)**(13/2)/(13*e**3) + 4*b*(d + e*x)**(11/2)*(a*e - b*d)/(11*e**3)
+ 2*(d + e*x)**(9/2)*(a*e - b*d)**2/(9*e**3)

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Mathematica [A]  time = 0.120076, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(9/2)*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^2*(8*d^2 - 36*d*e*
x + 99*e^2*x^2)))/(1287*e^3)

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Maple [A]  time = 0.011, size = 63, normalized size = 0.9 \[{\frac{198\,{x}^{2}{b}^{2}{e}^{2}+468\,xab{e}^{2}-72\,x{b}^{2}de+286\,{a}^{2}{e}^{2}-104\,abde+16\,{b}^{2}{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*(e*x+d)^(9/2)*(99*b^2*e^2*x^2+234*a*b*e^2*x-36*b^2*d*e*x+143*a^2*e^2-52*a
*b*d*e+8*b^2*d^2)/e^3

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Maxima [A]  time = 0.730934, size = 92, normalized size = 1.3 \[ \frac{2 \,{\left (99 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{2} - 234 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 143 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*b^2 - 234*(b^2*d - a*b*e)*(e*x + d)^(11/2) + 143*(b^
2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x + d)^(9/2))/e^3

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Fricas [A]  time = 0.204879, size = 286, normalized size = 4.03 \[ \frac{2 \,{\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \,{\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} +{\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \,{\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/1287*(99*b^2*e^6*x^6 + 8*b^2*d^6 - 52*a*b*d^5*e + 143*a^2*d^4*e^2 + 18*(20*b^2
*d*e^5 + 13*a*b*e^6)*x^5 + (458*b^2*d^2*e^4 + 884*a*b*d*e^5 + 143*a^2*e^6)*x^4 +
4*(53*b^2*d^3*e^3 + 299*a*b*d^2*e^4 + 143*a^2*d*e^5)*x^3 + 3*(b^2*d^4*e^2 + 208
*a*b*d^3*e^3 + 286*a^2*d^2*e^4)*x^2 - 2*(2*b^2*d^5*e - 13*a*b*d^4*e^2 - 286*a^2*
d^3*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 23.1524, size = 432, normalized size = 6.08 \[ \begin{cases} \frac{2 a^{2} d^{4} \sqrt{d + e x}}{9 e} + \frac{8 a^{2} d^{3} x \sqrt{d + e x}}{9} + \frac{4 a^{2} d^{2} e x^{2} \sqrt{d + e x}}{3} + \frac{8 a^{2} d e^{2} x^{3} \sqrt{d + e x}}{9} + \frac{2 a^{2} e^{3} x^{4} \sqrt{d + e x}}{9} - \frac{8 a b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{4 a b d^{4} x \sqrt{d + e x}}{99 e} + \frac{32 a b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{184 a b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{136 a b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{4 a b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 b^{2} d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 b^{2} d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 b^{2} d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 b^{2} d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 b^{2} d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 b^{2} d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 b^{2} e^{3} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*a**2*d**3*x*sqrt(d + e*x)/9 + 4*a
**2*d**2*e*x**2*sqrt(d + e*x)/3 + 8*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**2*e*
*3*x**4*sqrt(d + e*x)/9 - 8*a*b*d**5*sqrt(d + e*x)/(99*e**2) + 4*a*b*d**4*x*sqrt
(d + e*x)/(99*e) + 32*a*b*d**3*x**2*sqrt(d + e*x)/33 + 184*a*b*d**2*e*x**3*sqrt(
d + e*x)/99 + 136*a*b*d*e**2*x**4*sqrt(d + e*x)/99 + 4*a*b*e**3*x**5*sqrt(d + e*
x)/11 + 16*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*b**2*d**5*x*sqrt(d + e*x)/(12
87*e**2) + 2*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*b**2*d**3*x**3*sqrt(d +
e*x)/1287 + 916*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*b**2*d*e**2*x**5*sqrt(d
+ e*x)/143 + 2*b**2*e**3*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(a**2*x +
a*b*x**2 + b**2*x**3/3), True))

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GIAC/XCAS [A]  time = 0.230349, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done