3.2472 \(\int \frac{(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=325 \[ \frac{3 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (B \left (-4 c e (a e+3 b d)+5 b^2 e^2+8 c^2 d^2\right )+4 A c e (2 c d-b e)\right )}{8 c^{7/2}}+\frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{e \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c (3 a B e+A b e+b B d)+8 A c^2 d+5 b^2 B e\right )+4 b c \left (-13 a B e^2+6 A c d e+4 B c d^2\right )-32 c^2 \left (-a A e^2-3 a B d e+A c d^2\right )-12 b^2 c e (A e+3 B d)+15 b^3 B e^2\right )}{4 c^3 \left (b^2-4 a c\right )} \]
[Out]
(2*(d + e*x)^2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*
e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (e*(15*b
^3*B*e^2 - 12*b^2*c*e*(3*B*d + A*e) - 32*c^2*(A*c*d^2 - 3*a*B*d*e - a*A*e^2) + 4
*b*c*(4*B*c*d^2 + 6*A*c*d*e - 13*a*B*e^2) - 2*c*e*(8*A*c^2*d + 5*b^2*B*e - 4*c*(
b*B*d + A*b*e + 3*a*B*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*e
*(4*A*c*e*(2*c*d - b*e) + B*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + a*e)))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))
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Rubi [A] time = 0.937173, antiderivative size = 325, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148
\[ \frac{3 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (B \left (-4 c e (a e+3 b d)+5 b^2 e^2+8 c^2 d^2\right )+4 A c e (2 c d-b e)\right )}{8 c^{7/2}}+\frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{e \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c (3 a B e+A b e+b B d)+8 A c^2 d+5 b^2 B e\right )+4 b c \left (-13 a B e^2+6 A c d e+4 B c d^2\right )-32 c^2 \left (-a A e^2-3 a B d e+A c d^2\right )-12 b^2 c e (A e+3 B d)+15 b^3 B e^2\right )}{4 c^3 \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(3/2),x]
[Out]
(2*(d + e*x)^2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*
e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (e*(15*b
^3*B*e^2 - 12*b^2*c*e*(3*B*d + A*e) - 32*c^2*(A*c*d^2 - 3*a*B*d*e - a*A*e^2) + 4
*b*c*(4*B*c*d^2 + 6*A*c*d*e - 13*a*B*e^2) - 2*c*e*(8*A*c^2*d + 5*b^2*B*e - 4*c*(
b*B*d + A*b*e + 3*a*B*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*e
*(4*A*c*e*(2*c*d - b*e) + B*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + a*e)))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))
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Rubi in Sympy [A] time = 120.351, size = 372, normalized size = 1.14 \[ - \frac{2 \left (d + e x\right )^{2} \left (- 2 a c \left (A e + B d\right ) + b \left (A c d + B a e\right ) - x \left (- 2 A c^{2} d - B b^{2} e + c \left (A b e + 2 B a e + B b d\right )\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{e \sqrt{a + b x + c x^{2}} \left (- 8 A a c^{2} e^{2} + 3 A b^{2} c e^{2} - 6 A b c^{2} d e + 8 A c^{3} d^{2} + 13 B a b c e^{2} - 24 B a c^{2} d e - \frac{15 B b^{3} e^{2}}{4} + 9 B b^{2} c d e - 4 B b c^{2} d^{2} + \frac{c e x \left (- 4 A b c e + 8 A c^{2} d - 12 B a c e + 5 B b^{2} e - 4 B b c d\right )}{2}\right )}{c^{3} \left (- 4 a c + b^{2}\right )} + \frac{3 e \left (- 4 A b c e^{2} + 8 A c^{2} d e - 4 B a c e^{2} + 5 B b^{2} e^{2} - 12 B b c d e + 8 B c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} \]
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)**(3/2),x)
[Out]
-2*(d + e*x)**2*(-2*a*c*(A*e + B*d) + b*(A*c*d + B*a*e) - x*(-2*A*c**2*d - B*b**
2*e + c*(A*b*e + 2*B*a*e + B*b*d)))/(c*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)) +
e*sqrt(a + b*x + c*x**2)*(-8*A*a*c**2*e**2 + 3*A*b**2*c*e**2 - 6*A*b*c**2*d*e +
8*A*c**3*d**2 + 13*B*a*b*c*e**2 - 24*B*a*c**2*d*e - 15*B*b**3*e**2/4 + 9*B*b**2
*c*d*e - 4*B*b*c**2*d**2 + c*e*x*(-4*A*b*c*e + 8*A*c**2*d - 12*B*a*c*e + 5*B*b**
2*e - 4*B*b*c*d)/2)/(c**3*(-4*a*c + b**2)) + 3*e*(-4*A*b*c*e**2 + 8*A*c**2*d*e -
4*B*a*c*e**2 + 5*B*b**2*e**2 - 12*B*b*c*d*e + 8*B*c**2*d**2)*atanh((b + 2*c*x)/
(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*c**(7/2))
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Mathematica [A] time = 1.05514, size = 401, normalized size = 1.23 \[ \frac{B \left (4 a^2 c e^2 (6 c (4 d+e x)-13 b e)+a \left (15 b^3 e^3-2 b^2 c e^2 (18 d+31 e x)+4 b c^2 e \left (6 d^2+30 d e x-5 e^2 x^2\right )-8 c^3 \left (2 d^3+6 d^2 e x-6 d e^2 x^2-e^3 x^3\right )\right )+b x \left (15 b^3 e^3+b^2 c e^2 (5 e x-36 d)-2 b c^2 e \left (-12 d^2+6 d e x+e^2 x^2\right )-8 c^3 d^3\right )\right )-4 A c \left (-4 c \left (2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x\right )+b^2 e^2 (3 a e+c x (e x-6 d))-2 b c \left (a e^2 (3 d+5 e x)+c d^2 (d-3 e x)\right )+3 b^3 e^3 x\right )}{4 c^3 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{3 e \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (B \left (-4 c e (a e+3 b d)+5 b^2 e^2+8 c^2 d^2\right )+4 A c e (2 c d-b e)\right )}{8 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(3/2),x]
[Out]
(-4*A*c*(3*b^3*e^3*x + b^2*e^2*(3*a*e + c*x*(-6*d + e*x)) - 2*b*c*(c*d^2*(d - 3*
e*x) + a*e^2*(3*d + 5*e*x)) - 4*c*(2*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e
*x + e^2*x^2))) + B*(4*a^2*c*e^2*(-13*b*e + 6*c*(4*d + e*x)) + b*x*(-8*c^3*d^3 +
15*b^3*e^3 + b^2*c*e^2*(-36*d + 5*e*x) - 2*b*c^2*e*(-12*d^2 + 6*d*e*x + e^2*x^2
)) + a*(15*b^3*e^3 - 2*b^2*c*e^2*(18*d + 31*e*x) + 4*b*c^2*e*(6*d^2 + 30*d*e*x -
5*e^2*x^2) - 8*c^3*(2*d^3 + 6*d^2*e*x - 6*d*e^2*x^2 - e^3*x^3))))/(4*c^3*(-b^2
+ 4*a*c)*Sqrt[a + x*(b + c*x)]) + (3*e*(4*A*c*e*(2*c*d - b*e) + B*(8*c^2*d^2 + 5
*b^2*e^2 - 4*c*e*(3*b*d + a*e)))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]
])/(8*c^(7/2))
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Maple [B] time = 0.023, size = 1451, normalized size = 4.5 \[ \text{result too large to display} \]
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)^(3/2),x)
[Out]
3/2/c^2*b/(c*x^2+b*x+a)^(1/2)*A*d*e^2+3/2/c^2*b/(c*x^2+b*x+a)^(1/2)*B*d^2*e+3*x^
2/c/(c*x^2+b*x+a)^(1/2)*B*d*e^2+3/2/c^2*b*x/(c*x^2+b*x+a)^(1/2)*A*e^3-9/4/c^3*b^
2/(c*x^2+b*x+a)^(1/2)*B*d*e^2-3/4/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*e^3-
9/2/c^(5/2)*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*e^2+6*a/c^2/(c*x^2
+b*x+a)^(1/2)*B*d*e^2-3*x/c/(c*x^2+b*x+a)^(1/2)*A*d*e^2-15/8*B*e^3/c^3*b^2*x/(c*
x^2+b*x+a)^(1/2)+15/16*B*e^3/c^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-13/4*B*e^3/
c^3*b*a/(c*x^2+b*x+a)^(1/2)+3/2*B*e^3*a/c^2*x/(c*x^2+b*x+a)^(1/2)-3*x/c/(c*x^2+b
*x+a)^(1/2)*B*d^2*e-2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d^3-b^2/c/(4*a*c-b^2
)/(c*x^2+b*x+a)^(1/2)*B*d^3-5/4*B*e^3/c^2*b*x^2/(c*x^2+b*x+a)^(1/2)+12*a/c*b/(4*
a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d*e^2-1/c/(c*x^2+b*x+a)^(1/2)*B*d^3-9/2/c^2*b^3
/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d*e^2+4*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)*x*A*e^3+6*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*B*d*e^2+3/c*b^2/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*x*A*d*e^2+3/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d^2
*e-13/2*B*e^3/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/4/c^3*b^2/(c*x^2+b*x
+a)^(1/2)*A*e^3-3/2/c^(5/2)*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*e^3+
2*a/c^2/(c*x^2+b*x+a)^(1/2)*A*e^3+2*A*d^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)-3/c/(c*x^2+b*x+a)^(1/2)*A*d^2*e+1/2*B*e^3*x^3/c/(c*x^2+b*x+a)^(1/2)+15/16*B*
e^3/c^4*b^3/(c*x^2+b*x+a)^(1/2)+3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*A*d*e^2+3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^2*e+x^2/
c/(c*x^2+b*x+a)^(1/2)*A*e^3-3/2*B*e^3*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))+15/8*B*e^3/c^(7/2)*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2
*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*e^3+3/2/c^2*b^3/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*A*d*e^2+3/2/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*B*d^2*e+15/8*B*
e^3/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-13/4*B*e^3/c^3*b^3*a/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)+9/2/c^2*b*x/(c*x^2+b*x+a)^(1/2)*B*d*e^2-3/2/c^2*b^3/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*x*A*e^3-6*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e-3*b
^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*d^2*e-9/4/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+
a)^(1/2)*B*d*e^2
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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)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.52136, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)^(3/2),x, algorithm="fricas")
[Out]
[1/16*(4*(8*(2*B*a - A*b)*c^3*d^3 + 2*(B*b^2*c^2 - 4*B*a*c^3)*e^3*x^3 - 24*(B*a*
b*c^2 - 2*A*a*c^3)*d^2*e + 12*(3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^2)*d*e^2 - (1
5*B*a*b^3 + 32*A*a^2*c^2 - 4*(13*B*a^2*b + 3*A*a*b^2)*c)*e^3 + (12*(B*b^2*c^2 -
4*B*a*c^3)*d*e^2 - (5*B*b^3*c + 16*A*a*c^3 - 4*(5*B*a*b + A*b^2)*c^2)*e^3)*x^2 +
(8*(B*b*c^3 - 2*A*c^4)*d^3 - 24*(B*b^2*c^2 - (2*B*a + A*b)*c^3)*d^2*e + 12*(3*B
*b^3*c + 4*A*a*c^3 - 2*(5*B*a*b + A*b^2)*c^2)*d*e^2 - (15*B*b^4 + 8*(3*B*a^2 + 5
*A*a*b)*c^2 - 2*(31*B*a*b^2 + 6*A*b^3)*c)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c)
- 3*(8*(B*a*b^2*c^2 - 4*B*a^2*c^3)*d^2*e - 4*(3*B*a*b^3*c + 8*A*a^2*c^3 - 2*(6*B
*a^2*b + A*a*b^2)*c^2)*d*e^2 + (5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 - 4*(6*B*a^
2*b^2 + A*a*b^3)*c)*e^3 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d^2*e - 4*(3*B*b^3*c^2 + 8*
A*a*c^4 - 2*(6*B*a*b + A*b^2)*c^3)*d*e^2 + (5*B*b^4*c + 16*(B*a^2 + A*a*b)*c^3 -
4*(6*B*a*b^2 + A*b^3)*c^2)*e^3)*x^2 + (8*(B*b^3*c^2 - 4*B*a*b*c^3)*d^2*e - 4*(3
*B*b^4*c + 8*A*a*b*c^3 - 2*(6*B*a*b^2 + A*b^3)*c^2)*d*e^2 + (5*B*b^5 + 16*(B*a^2
*b + A*a*b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)*e^3)*x)*log(4*(2*c^2*x + b*c)*sqrt(
c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/((a*b^2*c^3 - 4
*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(c)), 1/8*(2*(
8*(2*B*a - A*b)*c^3*d^3 + 2*(B*b^2*c^2 - 4*B*a*c^3)*e^3*x^3 - 24*(B*a*b*c^2 - 2*
A*a*c^3)*d^2*e + 12*(3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^2)*d*e^2 - (15*B*a*b^3
+ 32*A*a^2*c^2 - 4*(13*B*a^2*b + 3*A*a*b^2)*c)*e^3 + (12*(B*b^2*c^2 - 4*B*a*c^3)
*d*e^2 - (5*B*b^3*c + 16*A*a*c^3 - 4*(5*B*a*b + A*b^2)*c^2)*e^3)*x^2 + (8*(B*b*c
^3 - 2*A*c^4)*d^3 - 24*(B*b^2*c^2 - (2*B*a + A*b)*c^3)*d^2*e + 12*(3*B*b^3*c + 4
*A*a*c^3 - 2*(5*B*a*b + A*b^2)*c^2)*d*e^2 - (15*B*b^4 + 8*(3*B*a^2 + 5*A*a*b)*c^
2 - 2*(31*B*a*b^2 + 6*A*b^3)*c)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(8*(B
*a*b^2*c^2 - 4*B*a^2*c^3)*d^2*e - 4*(3*B*a*b^3*c + 8*A*a^2*c^3 - 2*(6*B*a^2*b +
A*a*b^2)*c^2)*d*e^2 + (5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 - 4*(6*B*a^2*b^2 + A
*a*b^3)*c)*e^3 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d^2*e - 4*(3*B*b^3*c^2 + 8*A*a*c^4 -
2*(6*B*a*b + A*b^2)*c^3)*d*e^2 + (5*B*b^4*c + 16*(B*a^2 + A*a*b)*c^3 - 4*(6*B*a
*b^2 + A*b^3)*c^2)*e^3)*x^2 + (8*(B*b^3*c^2 - 4*B*a*b*c^3)*d^2*e - 4*(3*B*b^4*c
+ 8*A*a*b*c^3 - 2*(6*B*a*b^2 + A*b^3)*c^2)*d*e^2 + (5*B*b^5 + 16*(B*a^2*b + A*a*
b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)*e^3)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqr
t(c*x^2 + b*x + a)*c)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3
*c^3 - 4*a*b*c^4)*x)*sqrt(-c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
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)**(3/2),x)
[Out]
Integral((A + B*x)*(d + e*x)**3/(a + b*x + c*x**2)**(3/2), x)
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GIAC/XCAS [A] time = 0.282284, size = 716, normalized size = 2.2 \[ \frac{{\left ({\left (\frac{2 \,{\left (B b^{2} c^{2} e^{3} - 4 \, B a c^{3} e^{3}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac{12 \, B b^{2} c^{2} d e^{2} - 48 \, B a c^{3} d e^{2} - 5 \, B b^{3} c e^{3} + 20 \, B a b c^{2} e^{3} + 4 \, A b^{2} c^{2} e^{3} - 16 \, A a c^{3} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac{8 \, B b c^{3} d^{3} - 16 \, A c^{4} d^{3} - 24 \, B b^{2} c^{2} d^{2} e + 48 \, B a c^{3} d^{2} e + 24 \, A b c^{3} d^{2} e + 36 \, B b^{3} c d e^{2} - 120 \, B a b c^{2} d e^{2} - 24 \, A b^{2} c^{2} d e^{2} + 48 \, A a c^{3} d e^{2} - 15 \, B b^{4} e^{3} + 62 \, B a b^{2} c e^{3} + 12 \, A b^{3} c e^{3} - 24 \, B a^{2} c^{2} e^{3} - 40 \, A a b c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac{16 \, B a c^{3} d^{3} - 8 \, A b c^{3} d^{3} - 24 \, B a b c^{2} d^{2} e + 48 \, A a c^{3} d^{2} e + 36 \, B a b^{2} c d e^{2} - 96 \, B a^{2} c^{2} d e^{2} - 24 \, A a b c^{2} d e^{2} - 15 \, B a b^{3} e^{3} + 52 \, B a^{2} b c e^{3} + 12 \, A a b^{2} c e^{3} - 32 \, A a^{2} c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (8 \, B c^{2} d^{2} e - 12 \, B b c d e^{2} + 8 \, A c^{2} d e^{2} + 5 \, B b^{2} e^{3} - 4 \, B a c e^{3} - 4 \, A b c e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{7}{2}}} \]
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)^(3/2),x, algorithm="giac")
[Out]
1/4*(((2*(B*b^2*c^2*e^3 - 4*B*a*c^3*e^3)*x/(b^2*c^3 - 4*a*c^4) + (12*B*b^2*c^2*d
*e^2 - 48*B*a*c^3*d*e^2 - 5*B*b^3*c*e^3 + 20*B*a*b*c^2*e^3 + 4*A*b^2*c^2*e^3 - 1
6*A*a*c^3*e^3)/(b^2*c^3 - 4*a*c^4))*x + (8*B*b*c^3*d^3 - 16*A*c^4*d^3 - 24*B*b^2
*c^2*d^2*e + 48*B*a*c^3*d^2*e + 24*A*b*c^3*d^2*e + 36*B*b^3*c*d*e^2 - 120*B*a*b*
c^2*d*e^2 - 24*A*b^2*c^2*d*e^2 + 48*A*a*c^3*d*e^2 - 15*B*b^4*e^3 + 62*B*a*b^2*c*
e^3 + 12*A*b^3*c*e^3 - 24*B*a^2*c^2*e^3 - 40*A*a*b*c^2*e^3)/(b^2*c^3 - 4*a*c^4))
*x + (16*B*a*c^3*d^3 - 8*A*b*c^3*d^3 - 24*B*a*b*c^2*d^2*e + 48*A*a*c^3*d^2*e + 3
6*B*a*b^2*c*d*e^2 - 96*B*a^2*c^2*d*e^2 - 24*A*a*b*c^2*d*e^2 - 15*B*a*b^3*e^3 + 5
2*B*a^2*b*c*e^3 + 12*A*a*b^2*c*e^3 - 32*A*a^2*c^2*e^3)/(b^2*c^3 - 4*a*c^4))/sqrt
(c*x^2 + b*x + a) - 3/8*(8*B*c^2*d^2*e - 12*B*b*c*d*e^2 + 8*A*c^2*d*e^2 + 5*B*b^
2*e^3 - 4*B*a*c*e^3 - 4*A*b*c*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*sqrt(c) - b))/c^(7/2)