3.2489 \(\int \frac{(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^3)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (4*(d + e*x)^2*(4*a*A*c*e + b^2*(4*B*d + 3*A*e) - 8*b*(A*c*d + a*B*
e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B*e))*x))/(15*(b^2 - 4*a*
c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(b^2*e*(5*B*d + 3*A*e) + 4*c*(4*A*c*d^2 + 3*
a*B*d*e + a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*(b*d - 2*a*e + (2*c*d
- b*e)*x))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.864346, antiderivative size = 264, 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 \[ -\frac{16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^3)/(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)^3)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (4*(d + e*x)^2*(4*a*A*c*e + b^2*(4*B*d + 3*A*e) - 8*b*(A*c*d + a*B*
e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B*e))*x))/(15*(b^2 - 4*a*
c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(b^2*e*(5*B*d + 3*A*e) + 4*c*(4*A*c*d^2 + 3*
a*B*d*e + a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*(b*d - 2*a*e + (2*c*d
- b*e)*x))/(15*(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)**3/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Mathematica [B]  time = 6.43767, size = 1119, normalized size = 4.24 \[ \frac{\left (c x^2+b x+a\right )^4 \left (\frac{2 \left (B e^3 x b^4+a B e^3 b^3-A c e^3 x b^3-3 B c d e^2 x b^3-a A c e^3 b^2-3 a B c d e^2 b^2-4 a B c e^3 x b^2+3 A c^2 d e^2 x b^2+3 B c^2 d^2 e x b^2+A c^3 d^3 b-3 a^2 B c e^3 b+3 a A c^2 d e^2 b+3 a B c^2 d^2 e b-B c^3 d^3 x b+3 a A c^2 e^3 x b+9 a B c^2 d e^2 x b-3 A c^3 d^2 e x b-2 a B c^3 d^3+2 a^2 A c^2 e^3+6 a^2 B c^2 d e^2-6 a A c^3 d^2 e+2 A c^4 d^3 x+2 a^2 B c^2 e^3 x-6 a A c^3 d e^2 x-6 a B c^3 d^2 e x\right )}{5 c^3 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )^3}+\frac{2 \left (-B e^3 b^5-4 A c e^3 b^4-12 B c d e^2 b^4-2 B c e^3 x b^4+24 a B c e^3 b^3+72 A c^2 d e^2 b^3+72 B c^2 d^2 e b^3-8 A c^2 e^3 x b^3-24 B c^2 d e^2 x b^3-64 B c^3 d^3 b^2-48 a A c^2 e^3 b^2-144 a B c^2 d e^2 b^2-192 A c^3 d^2 e b^2+48 a B c^2 e^3 x b^2+144 A c^3 d e^2 x b^2+144 B c^3 d^2 e x b^2+128 A c^4 d^3 b+48 a^2 B c^2 e^3 b+96 a A c^3 d e^2 b+96 a B c^3 d^2 e b-128 B c^4 d^3 x b-96 a A c^3 e^3 x b-288 a B c^3 d e^2 x b-384 A c^4 d^2 e x b+256 A c^5 d^3 x+96 a^2 B c^3 e^3 x+192 a A c^4 d e^2 x+192 a B c^4 d^2 e x\right )}{15 c^2 \left (4 a c-b^2\right )^3 \left (c x^2+b x+a\right )}+\frac{2 \left (3 B e^3 b^5-3 A c e^3 b^4-9 B c d e^2 b^4-4 B c e^3 x b^4-22 a B c e^3 b^3+9 A c^2 d e^2 b^3+9 B c^2 d^2 e b^3-A c^2 e^3 x b^3-3 B c^2 d e^2 x b^3-8 B c^3 d^3 b^2+14 a A c^2 e^3 b^2+42 a B c^2 d e^2 b^2-24 A c^3 d^2 e b^2+36 a B c^2 e^3 x b^2+18 A c^3 d e^2 x b^2+18 B c^3 d^2 e x b^2+16 A c^4 d^3 b+56 a^2 B c^2 e^3 b+12 a A c^3 d e^2 b+12 a B c^3 d^2 e b-16 B c^4 d^3 x b-12 a A c^3 e^3 x b-36 a B c^3 d e^2 x b-48 A c^4 d^2 e x b-40 a^2 A c^3 e^3-120 a^2 B c^3 d e^2+32 A c^5 d^3 x-48 a^2 B c^3 e^3 x+24 a A c^4 d e^2 x+24 a B c^4 d^2 e x\right )}{15 c^3 \left (4 a c-b^2\right )^2 \left (c x^2+b x+a\right )^2}\right )}{(a+x (b+c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((a + b*x + c*x^2)^4*((2*(A*b*c^3*d^3 - 2*a*B*c^3*d^3 + 3*a*b*B*c^2*d^2*e - 6*a*
A*c^3*d^2*e - 3*a*b^2*B*c*d*e^2 + 3*a*A*b*c^2*d*e^2 + 6*a^2*B*c^2*d*e^2 + a*b^3*
B*e^3 - a*A*b^2*c*e^3 - 3*a^2*b*B*c*e^3 + 2*a^2*A*c^2*e^3 - b*B*c^3*d^3*x + 2*A*
c^4*d^3*x + 3*b^2*B*c^2*d^2*e*x - 3*A*b*c^3*d^2*e*x - 6*a*B*c^3*d^2*e*x - 3*b^3*
B*c*d*e^2*x + 3*A*b^2*c^2*d*e^2*x + 9*a*b*B*c^2*d*e^2*x - 6*a*A*c^3*d*e^2*x + b^
4*B*e^3*x - A*b^3*c*e^3*x - 4*a*b^2*B*c*e^3*x + 3*a*A*b*c^2*e^3*x + 2*a^2*B*c^2*
e^3*x))/(5*c^3*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^3) + (2*(-8*b^2*B*c^3*d^3 + 16*A
*b*c^4*d^3 + 9*b^3*B*c^2*d^2*e - 24*A*b^2*c^3*d^2*e + 12*a*b*B*c^3*d^2*e - 9*b^4
*B*c*d*e^2 + 9*A*b^3*c^2*d*e^2 + 42*a*b^2*B*c^2*d*e^2 + 12*a*A*b*c^3*d*e^2 - 120
*a^2*B*c^3*d*e^2 + 3*b^5*B*e^3 - 3*A*b^4*c*e^3 - 22*a*b^3*B*c*e^3 + 14*a*A*b^2*c
^2*e^3 + 56*a^2*b*B*c^2*e^3 - 40*a^2*A*c^3*e^3 - 16*b*B*c^4*d^3*x + 32*A*c^5*d^3
*x + 18*b^2*B*c^3*d^2*e*x - 48*A*b*c^4*d^2*e*x + 24*a*B*c^4*d^2*e*x - 3*b^3*B*c^
2*d*e^2*x + 18*A*b^2*c^3*d*e^2*x - 36*a*b*B*c^3*d*e^2*x + 24*a*A*c^4*d*e^2*x - 4
*b^4*B*c*e^3*x - A*b^3*c^2*e^3*x + 36*a*b^2*B*c^2*e^3*x - 12*a*A*b*c^3*e^3*x - 4
8*a^2*B*c^3*e^3*x))/(15*c^3*(-b^2 + 4*a*c)^2*(a + b*x + c*x^2)^2) + (2*(-64*b^2*
B*c^3*d^3 + 128*A*b*c^4*d^3 + 72*b^3*B*c^2*d^2*e - 192*A*b^2*c^3*d^2*e + 96*a*b*
B*c^3*d^2*e - 12*b^4*B*c*d*e^2 + 72*A*b^3*c^2*d*e^2 - 144*a*b^2*B*c^2*d*e^2 + 96
*a*A*b*c^3*d*e^2 - b^5*B*e^3 - 4*A*b^4*c*e^3 + 24*a*b^3*B*c*e^3 - 48*a*A*b^2*c^2
*e^3 + 48*a^2*b*B*c^2*e^3 - 128*b*B*c^4*d^3*x + 256*A*c^5*d^3*x + 144*b^2*B*c^3*
d^2*e*x - 384*A*b*c^4*d^2*e*x + 192*a*B*c^4*d^2*e*x - 24*b^3*B*c^2*d*e^2*x + 144
*A*b^2*c^3*d*e^2*x - 288*a*b*B*c^3*d*e^2*x + 192*a*A*c^4*d*e^2*x - 2*b^4*B*c*e^3
*x - 8*A*b^3*c^2*e^3*x + 48*a*b^2*B*c^2*e^3*x - 96*a*A*b*c^3*e^3*x + 96*a^2*B*c^
3*e^3*x))/(15*c^2*(-b^2 + 4*a*c)^3*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(7/2)

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Maple [B]  time = 0.021, size = 1502, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x)

[Out]

-2/15/(c*x^2+b*x+a)^(5/2)*(96*A*a*b*c^3*e^3*x^5-192*A*a*c^4*d*e^2*x^5+8*A*b^3*c^
2*e^3*x^5-144*A*b^2*c^3*d*e^2*x^5+384*A*b*c^4*d^2*e*x^5-256*A*c^5*d^3*x^5-96*B*a
^2*c^3*e^3*x^5-48*B*a*b^2*c^2*e^3*x^5+288*B*a*b*c^3*d*e^2*x^5-192*B*a*c^4*d^2*e*
x^5+2*B*b^4*c*e^3*x^5+24*B*b^3*c^2*d*e^2*x^5-144*B*b^2*c^3*d^2*e*x^5+128*B*b*c^4
*d^3*x^5+240*A*a*b^2*c^2*e^3*x^4-480*A*a*b*c^3*d*e^2*x^4+20*A*b^4*c*e^3*x^4-360*
A*b^3*c^2*d*e^2*x^4+960*A*b^2*c^3*d^2*e*x^4-640*A*b*c^4*d^3*x^4-240*B*a^2*b*c^2*
e^3*x^4-120*B*a*b^3*c*e^3*x^4+720*B*a*b^2*c^2*d*e^2*x^4-480*B*a*b*c^3*d^2*e*x^4+
5*B*b^5*e^3*x^4+60*B*b^4*c*d*e^2*x^4-360*B*b^3*c^2*d^2*e*x^4+320*B*b^2*c^3*d^3*x
^4+240*A*a^2*b*c^2*e^3*x^3-480*A*a^2*c^3*d*e^2*x^3+200*A*a*b^3*c*e^3*x^3-720*A*a
*b^2*c^2*d*e^2*x^3+960*A*a*b*c^3*d^2*e*x^3-640*A*a*c^4*d^3*x^3+15*A*b^5*e^3*x^3-
270*A*b^4*c*d*e^2*x^3+720*A*b^3*c^2*d^2*e*x^3-480*A*b^2*c^3*d^3*x^3-480*B*a^2*b^
2*c*e^3*x^3+720*B*a^2*b*c^2*d*e^2*x^3-480*B*a^2*c^3*d^2*e*x^3-40*B*a*b^4*e^3*x^3
+600*B*a*b^3*c*d*e^2*x^3-720*B*a*b^2*c^2*d^2*e*x^3+320*B*a*b*c^3*d^3*x^3+45*B*b^
5*d*e^2*x^3-270*B*b^4*c*d^2*e*x^3+240*B*b^3*c^2*d^3*x^3+160*A*a^3*c^2*e^3*x^2+24
0*A*a^2*b^2*c*e^3*x^2-720*A*a^2*b*c^2*d*e^2*x^2+90*A*a*b^4*e^3*x^2-600*A*a*b^3*c
*d*e^2*x^2+1440*A*a*b^2*c^2*d^2*e*x^2-960*A*a*b*c^3*d^3*x^2-45*A*b^5*d*e^2*x^2+1
20*A*b^4*c*d^2*e*x^2-80*A*b^3*c^2*d^3*x^2-320*B*a^3*b*c*e^3*x^2+480*B*a^3*c^2*d*
e^2*x^2-240*B*a^2*b^3*e^3*x^2+720*B*a^2*b^2*c*d*e^2*x^2-720*B*a^2*b*c^2*d^2*e*x^
2+270*B*a*b^4*d*e^2*x^2-600*B*a*b^3*c*d^2*e*x^2+480*B*a*b^2*c^2*d^3*x^2-45*B*b^5
*d^2*e*x^2+40*B*b^4*c*d^3*x^2+160*A*a^3*b*c*e^3*x+120*A*a^2*b^3*e^3*x-720*A*a^2*
b^2*c*d*e^2*x+720*A*a^2*b*c^2*d^2*e*x-480*A*a^2*c^3*d^3*x-60*A*a*b^4*d*e^2*x+360
*A*a*b^3*c*d^2*e*x-240*A*a*b^2*c^2*d^3*x-15*A*b^5*d^2*e*x+10*A*b^4*c*d^3*x-320*B
*a^3*b^2*e^3*x+480*B*a^3*b*c*d*e^2*x+360*B*a^2*b^3*d*e^2*x-720*B*a^2*b^2*c*d^2*e
*x+240*B*a^2*b*c^2*d^3*x-60*B*a*b^4*d^2*e*x+120*B*a*b^3*c*d^3*x-5*B*b^5*d^3*x+64
*A*a^4*c*e^3+48*A*a^3*b^2*e^3-288*A*a^3*b*c*d*e^2+288*A*a^3*c^2*d^2*e-24*A*a^2*b
^3*d*e^2+144*A*a^2*b^2*c*d^2*e-240*A*a^2*b*c^2*d^3-6*A*a*b^4*d^2*e+40*A*a*b^3*c*
d^3-3*A*b^5*d^3-128*B*a^4*b*e^3+192*B*a^4*c*d*e^2+144*B*a^3*b^2*d*e^2-288*B*a^3*
b*c*d^2*e+96*B*a^3*c^2*d^3-24*B*a^2*b^3*d^2*e+48*B*a^2*b^2*c*d^3-2*B*a*b^4*d^3)/
(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)^3/(c*x^2 + b*x + a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.889, size = 1850, normalized size = 7.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/(c*x^2 + b*x + a)^(7/2),x, algorithm="fricas")

[Out]

2/15*(2*(64*(B*b*c^4 - 2*A*c^5)*d^3 - 24*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d^2
*e + 12*(B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*d*e^2 + (B*b^4*c - 48*
(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*e^3)*x^5 + 5*(64*(B*b^2*c^3 - 2
*A*b*c^4)*d^3 - 24*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d^2*e + 12*(B*b^4*c -
8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*d*e^2 + (B*b^5 - 48*(B*a^2*b - A*a*b^2
)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*e^3)*x^4 - (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^3 - 6*(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^2*e + 24*(6*B*a^3*b^2 - A*a^2*b
^3 + 4*(2*B*a^4 - 3*A*a^3*b)*c)*d*e^2 - 16*(8*B*a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c)
*e^3 + 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^3 - 6*(9*B*
b^4*c + 16*(B*a^2 - 2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*d^2*e + 3*(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)*d*e^2
- (8*B*a*b^4 - 3*A*b^5 - 48*A*a^2*b*c^2 + 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c)*e^3)*x
^3 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^3 - 3*(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^2*e + 3*(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)*d*e^2
- 2*(24*B*a^2*b^3 - 9*A*a*b^4 - 16*A*a^3*c^2 + 8*(4*B*a^3*b - 3*A*a^2*b^2)*c)*e^
3)*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^3 + 3*(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^2*e - 12*(6*B*a^2*b^3 - A*a*b^4 + 4*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d*e^2
+ 8*(8*B*a^3*b^2 - 3*A*a^2*b^3 - 4*A*a^3*b*c)*e^3)*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)**3/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278223, 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)^3/(c*x^2 + b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done