3.23 \(\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=336 \[ -\frac{A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{-a^3 d^3 D+a^2 b C d^3+a b^2 d \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (-\left (-5 A d^3+2 B c d^2-2 c^3 D\right )\right )}{b^2 d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3+b^3 \left (-\left (5 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right )\right )}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 (-d) D-a^2 b (C d-6 c D)-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \]

[Out]

(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d^3*D - b^3*(2*c^2*C*d - 2*B*c*d^2 + 5*A*
d^3 - 2*c^3*D))/(3*b^3*d^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (A - (a*(b^2*B - a*b
*C + a^2*D))/b^3)/((b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - (a^2*b*C*d^3 - a^3*d
^3*D + a*b^2*d*(4*c*C*d - 3*B*d^2 - 6*c^2*D) - b^3*(2*B*c*d^2 - 5*A*d^3 - 2*c^3*
D))/(b^2*d^2*(b*c - a*d)^3*Sqrt[c + d*x]) - ((b^3*(2*B*c - 5*A*d) - a*b^2*(4*c*C
- 3*B*d) - a^3*d*D - a^2*b*(C*d - 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 1.72581, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{-a^3 d^3 D+a^2 b C d^3+a b^2 d \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (-\left (-5 A d^3+2 B c d^2-2 c^3 D\right )\right )}{b^2 d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3+b^3 \left (-\left (5 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right )\right )}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 (-d) D-a^2 b (C d-6 c D)-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d^3*D - b^3*(2*c^2*C*d - 2*B*c*d^2 + 5*A*
d^3 - 2*c^3*D))/(3*b^3*d^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (A - (a*(b^2*B - a*b
*C + a^2*D))/b^3)/((b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - (a^2*b*C*d^3 - a^3*d
^3*D + a*b^2*d*(4*c*C*d - 3*B*d^2 - 6*c^2*D) - b^3*(2*B*c*d^2 - 5*A*d^3 - 2*c^3*
D))/(b^2*d^2*(b*c - a*d)^3*Sqrt[c + d*x]) - ((b^3*(2*B*c - 5*A*d) - a*b^2*(4*c*C
- 3*B*d) - a^3*d*D - a^2*b*(C*d - 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 173.146, size = 410, normalized size = 1.22 \[ - \frac{2 D}{b^{2} d^{2} \sqrt{c + d x}} + \frac{5 d \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{b^{2} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{2 \left (B b^{2} - 2 C a b + 3 D a^{2}\right )}{b^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{5 d \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{3 b^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{2 \left (B b^{2} - 2 C a b + 3 D a^{2}\right )}{3 b^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{A b^{3} - B a b^{2} + C a^{2} b - D a^{3}}{b^{3} \left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (C b d - 2 D a d - D b c\right )}{3 b^{3} d^{2} \left (c + d x\right )^{\frac{3}{2}}} + \frac{5 d \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{7}{2}}} + \frac{2 \left (B b^{2} - 2 C a b + 3 D a^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**2/(d*x+c)**(5/2),x)

[Out]

-2*D/(b**2*d**2*sqrt(c + d*x)) + 5*d*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(b*
*2*sqrt(c + d*x)*(a*d - b*c)**3) + 2*(B*b**2 - 2*C*a*b + 3*D*a**2)/(b**2*sqrt(c
+ d*x)*(a*d - b*c)**2) - 5*d*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(3*b**3*(c
+ d*x)**(3/2)*(a*d - b*c)**2) - 2*(B*b**2 - 2*C*a*b + 3*D*a**2)/(3*b**3*(c + d*x
)**(3/2)*(a*d - b*c)) + (A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(b**3*(a + b*x)*
(c + d*x)**(3/2)*(a*d - b*c)) - 2*(C*b*d - 2*D*a*d - D*b*c)/(3*b**3*d**2*(c + d*
x)**(3/2)) + 5*d*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)*atan(sqrt(b)*sqrt(c + d
*x)/sqrt(a*d - b*c))/(b**(3/2)*(a*d - b*c)**(7/2)) + 2*(B*b**2 - 2*C*a*b + 3*D*a
**2)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(b**(3/2)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 1.45345, size = 268, normalized size = 0.8 \[ \sqrt{c+d x} \left (\frac{a \left (a^2 D-a b C+b^2 B\right )-A b^3}{b (a+b x) (b c-a d)^3}+\frac{2 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{3 d^2 (c+d x)^2 (b c-a d)^2}+\frac{b \left (4 A d^3-2 B c d^2+2 c^3 D\right )-2 a d \left (B d^2+3 c^2 D-2 c C d\right )}{d^2 (c+d x) (a d-b c)^3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 (-d) D+a^2 b (6 c D-C d)+a b^2 (3 B d-4 c C)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

Sqrt[c + d*x]*((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))/(b*(b*c - a*d)^3*(a + b*x)
) + (2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(3*d^2*(b*c - a*d)^2*(c + d*x)^2)
+ (-2*a*d*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*(-2*B*c*d^2 + 4*A*d^3 + 2*c^3*D))/(d
^2*(-(b*c) + a*d)^3*(c + d*x))) - ((b^3*(2*B*c - 5*A*d) + a*b^2*(-4*c*C + 3*B*d)
- a^3*d*D + a^2*b*(-(C*d) + 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c -
a*d]])/(b^(3/2)*(b*c - a*d)^(7/2))

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Maple [B]  time = 0.038, size = 730, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((D*x^3+C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(5/2),x)

[Out]

-2/3*d/(a*d-b*c)^2/(d*x+c)^(3/2)*A+2/3/(a*d-b*c)^2/(d*x+c)^(3/2)*B*c-2/3/d/(a*d-
b*c)^2/(d*x+c)^(3/2)*C*c^2+2/3/d^2/(a*d-b*c)^2/(d*x+c)^(3/2)*D*c^3+4*d/(a*d-b*c)
^3/(d*x+c)^(1/2)*A*b-2*d/(a*d-b*c)^3/(d*x+c)^(1/2)*B*a-2/(a*d-b*c)^3/(d*x+c)^(1/
2)*B*b*c+4/(a*d-b*c)^3/(d*x+c)^(1/2)*C*a*c-6/d/(a*d-b*c)^3/(d*x+c)^(1/2)*D*a*c^2
+2/d^2/(a*d-b*c)^3/(d*x+c)^(1/2)*D*b*c^3+d/(a*d-b*c)^3*b^2*(d*x+c)^(1/2)/(b*d*x+
a*d)*A-d/(a*d-b*c)^3*b*(d*x+c)^(1/2)/(b*d*x+a*d)*B*a+d/(a*d-b*c)^3*(d*x+c)^(1/2)
/(b*d*x+a*d)*C*a^2-d/(a*d-b*c)^3/b*(d*x+c)^(1/2)/(b*d*x+a*d)*D*a^3+5*d/(a*d-b*c)
^3*b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*A-3*d/(a*
d-b*c)^3*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a-2
/(a*d-b*c)^3*b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))
*B*c+d/(a*d-b*c)^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2
))*C*a^2+4/(a*d-b*c)^3*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b
)^(1/2))*C*a*c+d/(a*d-b*c)^3/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-
b*c)*b)^(1/2))*a^3*D-6/(a*d-b*c)^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((
a*d-b*c)*b)^(1/2))*D*a^2*c

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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((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253691, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/6*(3*(2*(3*D*a^3*b - 2*C*a^2*b^2 + B*a*b^3)*c^2*d^2 - (D*a^4 + C*a^3*b - 3*B*
a^2*b^2 + 5*A*a*b^3)*c*d^3 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c*d^3 - (D*a^3
*b + C*a^2*b^2 - 3*B*a*b^3 + 5*A*b^4)*d^4)*x^2 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B
*b^4)*c^2*d^2 + 5*(D*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*c*d^3 - (D*a^4 + C*a^3
*b - 3*B*a^2*b^2 + 5*A*a*b^3)*d^4)*x)*sqrt(d*x + c)*log((sqrt(b^2*c - a*b*d)*(b*
d*x + 2*b*c - a*d) - 2*(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 2*(4*D*a*b^2*
c^4 - 2*A*a^2*b*d^4 - 2*(8*D*a^2*b - C*a*b^2)*c^3*d - (3*D*a^3 - 13*C*a^2*b + 11
*B*a*b^2 - 3*A*b^3)*c^2*d^2 - 2*(2*B*a^2*b - 7*A*a*b^2)*c*d^3 + 3*(2*D*b^3*c^3*d
- 6*D*a*b^2*c^2*d^2 + 2*(2*C*a*b^2 - B*b^3)*c*d^3 - (D*a^3 - C*a^2*b + 3*B*a*b^
2 - 5*A*b^3)*d^4)*x^2 + 2*(2*D*b^3*c^4 - (5*D*a*b^2 - C*b^3)*c^3*d - (9*D*a^2*b
- 5*C*a*b^2 + 4*B*b^3)*c^2*d^2 - (3*D*a^3 - 9*C*a^2*b + 8*B*a*b^2 - 10*A*b^3)*c*
d^3 - (3*B*a^2*b - 5*A*a*b^2)*d^4)*x)*sqrt(b^2*c - a*b*d))/((a*b^4*c^4*d^2 - 3*a
^2*b^3*c^3*d^3 + 3*a^3*b^2*c^2*d^4 - a^4*b*c*d^5 + (b^5*c^3*d^3 - 3*a*b^4*c^2*d^
4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)*x^2 + (b^5*c^4*d^2 - 2*a*b^4*c^3*d^3 + 2*a^3*
b^2*c*d^5 - a^4*b*d^6)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c)), -1/3*(3*(2*(3*D*a^
3*b - 2*C*a^2*b^2 + B*a*b^3)*c^2*d^2 - (D*a^4 + C*a^3*b - 3*B*a^2*b^2 + 5*A*a*b^
3)*c*d^3 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c*d^3 - (D*a^3*b + C*a^2*b^2 - 3
*B*a*b^3 + 5*A*b^4)*d^4)*x^2 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c^2*d^2 + 5*
(D*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*c*d^3 - (D*a^4 + C*a^3*b - 3*B*a^2*b^2 +
5*A*a*b^3)*d^4)*x)*sqrt(d*x + c)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt
(d*x + c))) + (4*D*a*b^2*c^4 - 2*A*a^2*b*d^4 - 2*(8*D*a^2*b - C*a*b^2)*c^3*d - (
3*D*a^3 - 13*C*a^2*b + 11*B*a*b^2 - 3*A*b^3)*c^2*d^2 - 2*(2*B*a^2*b - 7*A*a*b^2)
*c*d^3 + 3*(2*D*b^3*c^3*d - 6*D*a*b^2*c^2*d^2 + 2*(2*C*a*b^2 - B*b^3)*c*d^3 - (D
*a^3 - C*a^2*b + 3*B*a*b^2 - 5*A*b^3)*d^4)*x^2 + 2*(2*D*b^3*c^4 - (5*D*a*b^2 - C
*b^3)*c^3*d - (9*D*a^2*b - 5*C*a*b^2 + 4*B*b^3)*c^2*d^2 - (3*D*a^3 - 9*C*a^2*b +
8*B*a*b^2 - 10*A*b^3)*c*d^3 - (3*B*a^2*b - 5*A*a*b^2)*d^4)*x)*sqrt(-b^2*c + a*b
*d))/((a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 3*a^3*b^2*c^2*d^4 - a^4*b*c*d^5 + (b^
5*c^3*d^3 - 3*a*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)*x^2 + (b^5*c^4*d^2
- 2*a*b^4*c^3*d^3 + 2*a^3*b^2*c*d^5 - a^4*b*d^6)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*
x + c))]

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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((D*x**3+C*x**2+B*x+A)/(b*x+a)**2/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230075, size = 593, normalized size = 1.76 \[ \frac{{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - D a^{3} d - C a^{2} b d + 3 \, B a b^{2} d - 5 \, A b^{3} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x + c} D a^{3} d - \sqrt{d x + c} C a^{2} b d + \sqrt{d x + c} B a b^{2} d - \sqrt{d x + c} A b^{3} d}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac{2 \,{\left (3 \,{\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \,{\left (d x + c\right )} D a c^{2} d + D a c^{3} d + C b c^{3} d + 6 \,{\left (d x + c\right )} C a c d^{2} - 3 \,{\left (d x + c\right )} B b c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \,{\left (d x + c\right )} B a d^{3} + 6 \,{\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \,{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

(6*D*a^2*b*c - 4*C*a*b^2*c + 2*B*b^3*c - D*a^3*d - C*a^2*b*d + 3*B*a*b^2*d - 5*A
*b^3*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^3 - 3*a*b^3*c^2*d +
3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(-b^2*c + a*b*d)) + (sqrt(d*x + c)*D*a^3*d - s
qrt(d*x + c)*C*a^2*b*d + sqrt(d*x + c)*B*a*b^2*d - sqrt(d*x + c)*A*b^3*d)/((b^4*
c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*((d*x + c)*b - b*c + a*d)) -
2/3*(3*(d*x + c)*D*b*c^3 - D*b*c^4 - 9*(d*x + c)*D*a*c^2*d + D*a*c^3*d + C*b*c^3
*d + 6*(d*x + c)*C*a*c*d^2 - 3*(d*x + c)*B*b*c*d^2 - C*a*c^2*d^2 - B*b*c^2*d^2 -
3*(d*x + c)*B*a*d^3 + 6*(d*x + c)*A*b*d^3 + B*a*c*d^3 + A*b*c*d^3 - A*a*d^4)/((
b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*(d*x + c)^(3/2))