3.1701 \(\int (a+b x) (A+B x) (d+e x)^{7/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac{2 b B (d+e x)^{13/2}}{13 e^3} \]

[Out]

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

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Rubi [A]  time = 0.119191, antiderivative size = 83, 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 \[ -\frac{2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac{2 b B (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.713, size = 78, normalized size = 0.94 \[ \frac{2 B b \left (d + e x\right )^{\frac{13}{2}}}{13 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (A b e + B a e - 2 B b d\right )}{11 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (a e - b d\right )}{9 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.151067, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (13 a e (11 A e-2 B d+9 B e x)+13 A b e (9 e x-2 d)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(9/2)*(13*A*b*e*(-2*d + 9*e*x) + 13*a*e*(-2*B*d + 11*A*e + 9*B*e*x)
+ b*B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)

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Maple [A]  time = 0.007, size = 73, normalized size = 0.9 \[{\frac{198\,bB{x}^{2}{e}^{2}+234\,Ab{e}^{2}x+234\,Ba{e}^{2}x-72\,Bbdex+286\,aA{e}^{2}-52\,Abde-52\,Bade+16\,bB{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*(e*x+d)^(9/2)*(99*B*b*e^2*x^2+117*A*b*e^2*x+117*B*a*e^2*x-36*B*b*d*e*x+14
3*A*a*e^2-26*A*b*d*e-26*B*a*d*e+8*B*b*d^2)/e^3

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Maxima [A]  time = 1.34898, size = 101, normalized size = 1.22 \[ \frac{2 \,{\left (99 \,{\left (e x + d\right )}^{\frac{13}{2}} B b - 117 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 143 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222775, size = 311, normalized size = 3.75 \[ \frac{2 \,{\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \,{\left (B a + A b\right )} d^{5} e + 9 \,{\left (40 \, B b d e^{5} + 13 \,{\left (B a + A b\right )} e^{6}\right )} x^{5} +{\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \,{\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \,{\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \,{\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \,{\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \,{\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} -{\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \,{\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*(99*B*b*e^6*x^6 + 8*B*b*d^6 + 143*A*a*d^4*e^2 - 26*(B*a + A*b)*d^5*e + 9*
(40*B*b*d*e^5 + 13*(B*a + A*b)*e^6)*x^5 + (458*B*b*d^2*e^4 + 143*A*a*e^6 + 442*(
B*a + A*b)*d*e^5)*x^4 + 2*(106*B*b*d^3*e^3 + 286*A*a*d*e^5 + 299*(B*a + A*b)*d^2
*e^4)*x^3 + 3*(B*b*d^4*e^2 + 286*A*a*d^2*e^4 + 104*(B*a + A*b)*d^3*e^3)*x^2 - (4
*B*b*d^5*e - 572*A*a*d^3*e^3 - 13*(B*a + A*b)*d^4*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 25.2017, size = 578, normalized size = 6.96 \[ \begin{cases} \frac{2 A a d^{4} \sqrt{d + e x}}{9 e} + \frac{8 A a d^{3} x \sqrt{d + e x}}{9} + \frac{4 A a d^{2} e x^{2} \sqrt{d + e x}}{3} + \frac{8 A a d e^{2} x^{3} \sqrt{d + e x}}{9} + \frac{2 A a e^{3} x^{4} \sqrt{d + e x}}{9} - \frac{4 A b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 A b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 A b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 A b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 A b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 A b e^{3} x^{5} \sqrt{d + e x}}{11} - \frac{4 B a d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 B a d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 B a d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 B a d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 B a d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 B a e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 B b d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 B b d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 B b d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 B b d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 B b d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 B b d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 B b e^{3} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*A*a*d**4*sqrt(d + e*x)/(9*e) + 8*A*a*d**3*x*sqrt(d + e*x)/9 + 4*A*a
*d**2*e*x**2*sqrt(d + e*x)/3 + 8*A*a*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a*e**3*x*
*4*sqrt(d + e*x)/9 - 4*A*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*A*b*d**4*x*sqrt(d +
e*x)/(99*e) + 16*A*b*d**3*x**2*sqrt(d + e*x)/33 + 92*A*b*d**2*e*x**3*sqrt(d + e*
x)/99 + 68*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d + e*x)/11 -
4*B*a*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a
*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a*d*e**
2*x**4*sqrt(d + e*x)/99 + 2*B*a*e**3*x**5*sqrt(d + e*x)/11 + 16*B*b*d**6*sqrt(d
+ e*x)/(1287*e**3) - 8*B*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*B*b*d**4*x**2*sq
rt(d + e*x)/(429*e) + 424*B*b*d**3*x**3*sqrt(d + e*x)/1287 + 916*B*b*d**2*e*x**4
*sqrt(d + e*x)/1287 + 80*B*b*d*e**2*x**5*sqrt(d + e*x)/143 + 2*B*b*e**3*x**6*sqr
t(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/
3), True))

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GIAC/XCAS [A]  time = 0.226669, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done