3.1784 \(\int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3*b*B*d
- 2*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e
)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(15*e^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.203316, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3*b*B*d
- 2*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e
)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(15*e^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 56.3922, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{13}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{13 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{11 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{9 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*b**2*(d + e*x)**(15/2)/(15*e**4) + 2*b*(d + e*x)**(13/2)*(A*b*e + 2*B*a*e -
3*B*b*d)/(13*e**4) + 2*(d + e*x)**(11/2)*(a*e - b*d)*(2*A*b*e + B*a*e - 3*B*b*d)
/(11*e**4) + 2*(d + e*x)**(9/2)*(A*e - B*d)*(a*e - b*d)**2/(9*e**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.296612, size = 138, normalized size = 1.08 \[ \frac{2 (d+e x)^{9/2} \left (65 a^2 e^2 (11 A e-2 B d+9 B e x)+10 a b e \left (13 A e (9 e x-2 d)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+b^2 \left (5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(9/2)*(65*a^2*e^2*(-2*B*d + 11*A*e + 9*B*e*x) + 10*a*b*e*(13*A*e*(-
2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) + b^2*(5*A*e*(8*d^2 - 36*d*e*x
+ 99*e^2*x^2) + B*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3))))/(6435
*e^4)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 169, normalized size = 1.3 \[{\frac{858\,B{x}^{3}{b}^{2}{e}^{3}+990\,A{b}^{2}{e}^{3}{x}^{2}+1980\,Bab{e}^{3}{x}^{2}-396\,B{b}^{2}d{e}^{2}{x}^{2}+2340\,Axab{e}^{3}-360\,Ax{b}^{2}d{e}^{2}+1170\,Bx{a}^{2}{e}^{3}-720\,Bxabd{e}^{2}+144\,B{b}^{2}{d}^{2}ex+1430\,A{a}^{2}{e}^{3}-520\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-260\,Bd{e}^{2}{a}^{2}+160\,B{d}^{2}abe-32\,B{b}^{2}{d}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/6435*(e*x+d)^(9/2)*(429*B*b^2*e^3*x^3+495*A*b^2*e^3*x^2+990*B*a*b*e^3*x^2-198*
B*b^2*d*e^2*x^2+1170*A*a*b*e^3*x-180*A*b^2*d*e^2*x+585*B*a^2*e^3*x-360*B*a*b*d*e
^2*x+72*B*b^2*d^2*e*x+715*A*a^2*e^3-260*A*a*b*d*e^2+40*A*b^2*d^2*e-130*B*a^2*d*e
^2+80*B*a*b*d^2*e-16*B*b^2*d^3)/e^4

_______________________________________________________________________________________

Maxima [A]  time = 0.746545, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{2} - 495 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 585 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/6435*(429*(e*x + d)^(15/2)*B*b^2 - 495*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x
+ d)^(13/2) + 585*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2
)*(e*x + d)^(11/2) - 715*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b^2)*d^2*e + (B*a
^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(9/2))/e^4

_______________________________________________________________________________________

Fricas [A]  time = 0.286503, size = 572, normalized size = 4.47 \[ \frac{2 \,{\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \,{\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \,{\left (46 \, B b^{2} d e^{6} + 15 \,{\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \,{\left (206 \, B b^{2} d^{2} e^{5} + 200 \,{\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \,{\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \,{\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \,{\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/6435*(429*B*b^2*e^7*x^7 - 16*B*b^2*d^7 + 715*A*a^2*d^4*e^3 + 40*(2*B*a*b + A*b
^2)*d^6*e - 130*(B*a^2 + 2*A*a*b)*d^5*e^2 + 33*(46*B*b^2*d*e^6 + 15*(2*B*a*b + A
*b^2)*e^7)*x^6 + 9*(206*B*b^2*d^2*e^5 + 200*(2*B*a*b + A*b^2)*d*e^6 + 65*(B*a^2
+ 2*A*a*b)*e^7)*x^5 + 5*(160*B*b^2*d^3*e^4 + 143*A*a^2*e^7 + 458*(2*B*a*b + A*b^
2)*d^2*e^5 + 442*(B*a^2 + 2*A*a*b)*d*e^6)*x^4 + 5*(B*b^2*d^4*e^3 + 572*A*a^2*d*e
^6 + 212*(2*B*a*b + A*b^2)*d^3*e^4 + 598*(B*a^2 + 2*A*a*b)*d^2*e^5)*x^3 - 3*(2*B
*b^2*d^5*e^2 - 1430*A*a^2*d^2*e^5 - 5*(2*B*a*b + A*b^2)*d^4*e^3 - 520*(B*a^2 + 2
*A*a*b)*d^3*e^4)*x^2 + (8*B*b^2*d^6*e + 2860*A*a^2*d^3*e^4 - 20*(2*B*a*b + A*b^2
)*d^5*e^2 + 65*(B*a^2 + 2*A*a*b)*d^4*e^3)*x)*sqrt(e*x + d)/e^4

_______________________________________________________________________________________

Sympy [A]  time = 31.9632, size = 1020, normalized size = 7.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*A*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**2*d**3*x*sqrt(d + e*x)/9 +
4*A*a**2*d**2*e*x**2*sqrt(d + e*x)/3 + 8*A*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2
*A*a**2*e**3*x**4*sqrt(d + e*x)/9 - 8*A*a*b*d**5*sqrt(d + e*x)/(99*e**2) + 4*A*a
*b*d**4*x*sqrt(d + e*x)/(99*e) + 32*A*a*b*d**3*x**2*sqrt(d + e*x)/33 + 184*A*a*b
*d**2*e*x**3*sqrt(d + e*x)/99 + 136*A*a*b*d*e**2*x**4*sqrt(d + e*x)/99 + 4*A*a*b
*e**3*x**5*sqrt(d + e*x)/11 + 16*A*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**
2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) +
424*A*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/1
287 + 80*A*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)
/13 - 4*B*a**2*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**2*d**4*x*sqrt(d + e*x)/(99*
e) + 16*B*a**2*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a**2*d**2*e*x**3*sqrt(d + e*x)/
99 + 68*B*a**2*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**2*e**3*x**5*sqrt(d + e*x)/1
1 + 32*B*a*b*d**6*sqrt(d + e*x)/(1287*e**3) - 16*B*a*b*d**5*x*sqrt(d + e*x)/(128
7*e**2) + 4*B*a*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 848*B*a*b*d**3*x**3*sqrt(d +
e*x)/1287 + 1832*B*a*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 160*B*a*b*d*e**2*x**5*s
qrt(d + e*x)/143 + 4*B*a*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*b**2*d**7*sqrt(d +
e*x)/(6435*e**4) + 16*B*b**2*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x*
*2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*B
*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 +
92*B*b**2*d*e**2*x**6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15, N
e(e, 0)), (d**(7/2)*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B
*a*b*x**3/3 + B*b**2*x**4/4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.339503, 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)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done