3.1213 \(\int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac{2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac{2 B c (d+e x)^{15/2}}{15 e^4} \]

[Out]

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

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Rubi [A]  time = 0.2358, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac{2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac{2 B c (d+e x)^{15/2}}{15 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.8307, size = 128, normalized size = 1.02 \[ \frac{2 B c \left (d + e x\right )^{\frac{15}{2}}}{15 e^{4}} - \frac{2 d \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (b e - c d\right )}{9 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (A c e + B b e - 3 B c d\right )}{13 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{11 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.247368, size = 113, normalized size = 0.9 \[ \frac{2 (d+e x)^{9/2} \left (5 A e \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+B \left (5 b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+c \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 121, normalized size = 1. \[ -{\frac{-858\,Bc{x}^{3}{e}^{3}-990\,Ac{e}^{3}{x}^{2}-990\,Bb{e}^{3}{x}^{2}+396\,Bcd{e}^{2}{x}^{2}-1170\,Ab{e}^{3}x+360\,Acd{e}^{2}x+360\,Bbd{e}^{2}x-144\,Bc{d}^{2}ex+260\,Abd{e}^{2}-80\,Ac{d}^{2}e-80\,Bb{d}^{2}e+32\,Bc{d}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/6435*(e*x+d)^(9/2)*(-429*B*c*e^3*x^3-495*A*c*e^3*x^2-495*B*b*e^3*x^2+198*B*c*
d*e^2*x^2-585*A*b*e^3*x+180*A*c*d*e^2*x+180*B*b*d*e^2*x-72*B*c*d^2*e*x+130*A*b*d
*e^2-40*A*c*d^2*e-40*B*b*d^2*e+16*B*c*d^3)/e^4

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Maxima [A]  time = 0.699881, size = 151, normalized size = 1.2 \[ \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} B c - 495 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 585 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.285455, size = 366, normalized size = 2.9 \[ \frac{2 \,{\left (429 \, B c e^{7} x^{7} - 16 \, B c d^{7} - 130 \, A b d^{5} e^{2} + 40 \,{\left (B b + A c\right )} d^{6} e + 33 \,{\left (46 \, B c d e^{6} + 15 \,{\left (B b + A c\right )} e^{7}\right )} x^{6} + 9 \,{\left (206 \, B c d^{2} e^{5} + 65 \, A b e^{7} + 200 \,{\left (B b + A c\right )} d e^{6}\right )} x^{5} + 10 \,{\left (80 \, B c d^{3} e^{4} + 221 \, A b d e^{6} + 229 \,{\left (B b + A c\right )} d^{2} e^{5}\right )} x^{4} + 5 \,{\left (B c d^{4} e^{3} + 598 \, A b d^{2} e^{5} + 212 \,{\left (B b + A c\right )} d^{3} e^{4}\right )} x^{3} - 3 \,{\left (2 \, B c d^{5} e^{2} - 520 \, A b d^{3} e^{4} - 5 \,{\left (B b + A c\right )} d^{4} e^{3}\right )} x^{2} +{\left (8 \, B c d^{6} e + 65 \, A b d^{4} e^{3} - 20 \,{\left (B b + A c\right )} d^{5} e^{2}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/6435*(429*B*c*e^7*x^7 - 16*B*c*d^7 - 130*A*b*d^5*e^2 + 40*(B*b + A*c)*d^6*e +
33*(46*B*c*d*e^6 + 15*(B*b + A*c)*e^7)*x^6 + 9*(206*B*c*d^2*e^5 + 65*A*b*e^7 + 2
00*(B*b + A*c)*d*e^6)*x^5 + 10*(80*B*c*d^3*e^4 + 221*A*b*d*e^6 + 229*(B*b + A*c)
*d^2*e^5)*x^4 + 5*(B*c*d^4*e^3 + 598*A*b*d^2*e^5 + 212*(B*b + A*c)*d^3*e^4)*x^3
- 3*(2*B*c*d^5*e^2 - 520*A*b*d^3*e^4 - 5*(B*b + A*c)*d^4*e^3)*x^2 + (8*B*c*d^6*e
+ 65*A*b*d^4*e^3 - 20*(B*b + A*c)*d^5*e^2)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 29.6058, size = 683, normalized size = 5.42 \[ \begin{cases} - \frac{4 A b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 A b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 A b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 A b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 A b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 A b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 A c d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 A c d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 A c d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 A c d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 A c d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 A c d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 A c e^{3} x^{6} \sqrt{d + e x}}{13} + \frac{16 B b d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 B b d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 B b d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 B b d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 B b d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 B b d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 B b e^{3} x^{6} \sqrt{d + e x}}{13} - \frac{32 B c d^{7} \sqrt{d + e x}}{6435 e^{4}} + \frac{16 B c d^{6} x \sqrt{d + e x}}{6435 e^{3}} - \frac{4 B c d^{5} x^{2} \sqrt{d + e x}}{2145 e^{2}} + \frac{2 B c d^{4} x^{3} \sqrt{d + e x}}{1287 e} + \frac{320 B c d^{3} x^{4} \sqrt{d + e x}}{1287} + \frac{412 B c d^{2} e x^{5} \sqrt{d + e x}}{715} + \frac{92 B c d e^{2} x^{6} \sqrt{d + e x}}{195} + \frac{2 B c e^{3} x^{7} \sqrt{d + e x}}{15} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\frac{A b x^{2}}{2} + \frac{A c x^{3}}{3} + \frac{B b x^{3}}{3} + \frac{B c x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-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 + 6
8*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*c*d
**6*sqrt(d + e*x)/(1287*e**3) - 8*A*c*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*c*d
**4*x**2*sqrt(d + e*x)/(429*e) + 424*A*c*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*c*
d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*c*d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*c*e*
*3*x**6*sqrt(d + e*x)/13 + 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*sqrt(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*sqrt(d + e*x)/13 - 32*B*c*d**7*sqrt(d
+ e*x)/(6435*e**4) + 16*B*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*c*d**5*x**2*
sqrt(d + e*x)/(2145*e**2) + 2*B*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*B*c*d**
3*x**4*sqrt(d + e*x)/1287 + 412*B*c*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*c*d*e**
2*x**6*sqrt(d + e*x)/195 + 2*B*c*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)), (d**(7/2
)*(A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4), True))

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GIAC/XCAS [A]  time = 0.322848, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done