3.2490 \(\int \frac{(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)
Optimal. Leaf size=324 \[ -\frac{2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{8 \left (b^2 \left (a B e^2+A c d e+2 B c d^2\right )+x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (8*(4*a*c*e*(3*A*c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^
2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3*b^2*c*e*(B*d + A*e) +
4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d*e - a*A*e^2))*x))/(15*c*(b^2
- 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (8*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) -
8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2) + 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e
^2))*(b + 2*c*x))/(15*c*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])
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Rubi [A] time = 0.969532, antiderivative size = 324, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111
\[ -\frac{2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{8 \left (b^2 \left (a B e^2+A c d e+2 B c d^2\right )+x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(7/2),x]
[Out]
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (8*(4*a*c*e*(3*A*c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^
2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3*b^2*c*e*(B*d + A*e) +
4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d*e - a*A*e^2))*x))/(15*c*(b^2
- 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (8*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) -
8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2) + 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e
^2))*(b + 2*c*x))/(15*c*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**2/(c*x**2+b*x+a)**(7/2),x)
[Out]
Timed out
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Mathematica [A] time = 5.06259, size = 416, normalized size = 1.28 \[ -\frac{2 \left (-3 \left (b^2-4 a c\right )^2 \left (B \left (-2 a^2 c e^2+a \left (b^2 e^2-b c e (2 d+3 e x)+2 c^2 d (d+2 e x)\right )+b x (c d-b e)^2\right )-A c \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )\right )+\left (b^2-4 a c\right ) (a+x (b+c x)) \left (-8 c^2 \left (-5 a^2 B e^2+a c e x (A e+2 B d)+4 A c^2 d^2 x\right )+2 b^2 c \left (-7 a B e^2+A c e (8 d-3 e x)+2 B c d (2 d-3 e x)\right )-4 b c^2 \left (a A e^2+a B e (2 d-3 e x)+4 A c d (d-2 e x)-4 B c d^2 x\right )+b^3 c e (-3 A e-6 B d+B e x)+3 b^4 B e^2\right )-4 c (b+2 c x) (a+x (b+c x))^2 \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )\right )}{15 c^2 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(7/2),x]
[Out]
(-2*(-4*c*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) - 8*c^2*(4*A*c*d^2 + 2*a*B*d*e +
a*A*e^2) + 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e^2))*(b + 2*c*x)*(a + x*(b + c*
x))^2 + (b^2 - 4*a*c)*(a + x*(b + c*x))*(3*b^4*B*e^2 + b^3*c*e*(-6*B*d - 3*A*e +
B*e*x) - 8*c^2*(-5*a^2*B*e^2 + 4*A*c^2*d^2*x + a*c*e*(2*B*d + A*e)*x) + 2*b^2*c
*(-7*a*B*e^2 + 2*B*c*d*(2*d - 3*e*x) + A*c*e*(8*d - 3*e*x)) - 4*b*c^2*(a*A*e^2 -
4*B*c*d^2*x + a*B*e*(2*d - 3*e*x) + 4*A*c*d*(d - 2*e*x))) - 3*(b^2 - 4*a*c)^2*(
-(A*c*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*
x))) + B*(-2*a^2*c*e^2 + b*(c*d - b*e)^2*x + a*(b^2*e^2 + 2*c^2*d*(d + 2*e*x) -
b*c*e*(2*d + 3*e*x))))))/(15*c^2*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))
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Maple [B] time = 0.017, size = 1064, normalized size = 3.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),x)
[Out]
2/15/(c*x^2+b*x+a)^(5/2)*(64*A*a*c^4*e^2*x^5+48*A*b^2*c^3*e^2*x^5-256*A*b*c^4*d*
e*x^5+256*A*c^5*d^2*x^5-96*B*a*b*c^3*e^2*x^5+128*B*a*c^4*d*e*x^5-8*B*b^3*c^2*e^2
*x^5+96*B*b^2*c^3*d*e*x^5-128*B*b*c^4*d^2*x^5+160*A*a*b*c^3*e^2*x^4+120*A*b^3*c^
2*e^2*x^4-640*A*b^2*c^3*d*e*x^4+640*A*b*c^4*d^2*x^4-240*B*a*b^2*c^2*e^2*x^4+320*
B*a*b*c^3*d*e*x^4-20*B*b^4*c*e^2*x^4+240*B*b^3*c^2*d*e*x^4-320*B*b^2*c^3*d^2*x^4
+160*A*a^2*c^3*e^2*x^3+240*A*a*b^2*c^2*e^2*x^3-640*A*a*b*c^3*d*e*x^3+640*A*a*c^4
*d^2*x^3+90*A*b^4*c*e^2*x^3-480*A*b^3*c^2*d*e*x^3+480*A*b^2*c^3*d^2*x^3-240*B*a^
2*b*c^2*e^2*x^3+320*B*a^2*c^3*d*e*x^3-200*B*a*b^3*c*e^2*x^3+480*B*a*b^2*c^2*d*e*
x^3-320*B*a*b*c^3*d^2*x^3-15*B*b^5*e^2*x^3+180*B*b^4*c*d*e*x^3-240*B*b^3*c^2*d^2
*x^3+240*A*a^2*b*c^2*e^2*x^2+200*A*a*b^3*c*e^2*x^2-960*A*a*b^2*c^2*d*e*x^2+960*A
*a*b*c^3*d^2*x^2+15*A*b^5*e^2*x^2-80*A*b^4*c*d*e*x^2+80*A*b^3*c^2*d^2*x^2-160*B*
a^3*c^2*e^2*x^2-240*B*a^2*b^2*c*e^2*x^2+480*B*a^2*b*c^2*d*e*x^2-90*B*a*b^4*e^2*x
^2+400*B*a*b^3*c*d*e*x^2-480*B*a*b^2*c^2*d^2*x^2+30*B*b^5*d*e*x^2-40*B*b^4*c*d^2
*x^2+240*A*a^2*b^2*c*e^2*x-480*A*a^2*b*c^2*d*e*x+480*A*a^2*c^3*d^2*x+20*A*a*b^4*
e^2*x-240*A*a*b^3*c*d*e*x+240*A*a*b^2*c^2*d^2*x+10*A*b^5*d*e*x-10*A*b^4*c*d^2*x-
160*B*a^3*b*c*e^2*x-120*B*a^2*b^3*e^2*x+480*B*a^2*b^2*c*d*e*x-240*B*a^2*b*c^2*d^
2*x+40*B*a*b^4*d*e*x-120*B*a*b^3*c*d^2*x+5*B*b^5*d^2*x+96*A*a^3*b*c*e^2-192*A*a^
3*c^2*d*e+8*A*a^2*b^3*e^2-96*A*a^2*b^2*c*d*e+240*A*a^2*b*c^2*d^2+4*A*a*b^4*d*e-4
0*A*a*b^3*c*d^2+3*A*b^5*d^2-64*B*a^4*c*e^2-48*B*a^3*b^2*e^2+192*B*a^3*b*c*d*e-96
*B*a^3*c^2*d^2+16*B*a^2*b^3*d*e-48*B*a^2*b^2*c*d^2+2*B*a*b^4*d^2)/(64*a^3*c^3-48
*a^2*b^2*c^2+12*a*b^4*c-b^6)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(c*x^2 + b*x + a)^(7/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.83778, size = 1478, normalized size = 4.56 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(c*x^2 + b*x + a)^(7/2),x, algorithm="fricas")
[Out]
2/15*(8*(16*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d*e
+ (B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*e^2)*x^5 + 20*(16*(B*b^2*c^3
- 2*A*b*c^4)*d^2 - 4*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d*e + (B*b^4*c - 8
*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*e^2)*x^4 + 5*(16*(3*B*b^3*c^2 - 8*A*a*c^
4 + 2*(2*B*a*b - 3*A*b^2)*c^3)*d^2 - 4*(9*B*b^4*c + 16*(B*a^2 - 2*A*a*b)*c^3 + 2
4*(B*a*b^2 - A*b^3)*c^2)*d*e + (3*B*b^5 - 32*A*a^2*c^3 + 48*(B*a^2*b - A*a*b^2)*
c^2 + 2*(20*B*a*b^3 - 9*A*b^4)*c)*e^2)*x^3 - (2*B*a*b^4 + 3*A*b^5 - 48*(2*B*a^3
- 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 + 5*A*a*b^3)*c)*d^2 - 4*(4*B*a^2*b^3 + A*a*b^4
- 48*A*a^3*c^2 + 24*(2*B*a^3*b - A*a^2*b^2)*c)*d*e + 8*(6*B*a^3*b^2 - A*a^2*b^3
+ 4*(2*B*a^4 - 3*A*a^3*b)*c)*e^2 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2
- A*b^3)*c^2)*d^2 - 2*(3*B*b^5 + 48*(B*a^2*b - 2*A*a*b^2)*c^2 + 8*(5*B*a*b^3 - A
*b^4)*c)*d*e + (18*B*a*b^4 - 3*A*b^5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 + 8*(6*B*a^2
*b^2 - 5*A*a*b^3)*c)*e^2)*x^2 - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^2*b - A*a*b^2
)*c^2 - 2*(12*B*a*b^3 + A*b^4)*c)*d^2 + 2*(4*B*a*b^4 + A*b^5 - 48*A*a^2*b*c^2 +
24*(2*B*a^2*b^2 - A*a*b^3)*c)*d*e - 4*(6*B*a^2*b^3 - A*a*b^4 + 4*(2*B*a^3*b - 3*
A*a^2*b^2)*c)*e^2)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2
*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 +
3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*
a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c
- 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6
*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5
*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**2/(c*x**2+b*x+a)**(7/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.278505, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(c*x^2 + b*x + a)^(7/2),x, algorithm="giac")
[Out]
Done