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

Optimal. Leaf size=438 \[ -\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}+\frac{a^3 \left (-d^2\right ) D+a^2 b C d^2+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3+b^3 \left (-\left (7 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\sqrt{c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{4 b (a+b x) (b c-a d)^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}} \]

[Out]

-(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 7*A
*d^3 - 4*c^3*D))/(6*b^3*d*(b*c - a*d)^3*(c + d*x)^(3/2)) - (A*b^3 - a*(b^2*B - a
*b*C + a^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*(c + d*x)^(3/2)) + (a^2*b*C*d^2 +
b^3*(2*c^2*C - 4*B*c*d + 7*A*d^2) - a^3*d^2*D + a*b^2*(4*c*C*d - 3*B*d^2 - 6*c^2
*D))/(b^2*(b*c - a*d)^4*Sqrt[c + d*x]) - ((b^3*(4*B*c - 7*A*d) - a*b^2*(8*c*C -
3*B*d) - 5*a^3*d*D + a^2*b*(C*d + 12*c*D))*Sqrt[c + d*x])/(4*b*(b*c - a*d)^4*(a
+ b*x)) - ((b^3*(8*c^2*C - 20*B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4
*c*D) + 3*a*b^2*(8*c*C*d - 5*B*d^2 - 8*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/S
qrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(9/2))

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Rubi [A]  time = 3.07157, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ -\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}+\frac{a^3 \left (-d^2\right ) D+a^2 b C d^2+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3+b^3 \left (-\left (7 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\sqrt{c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{4 b (a+b x) (b c-a d)^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(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*(4*c^2*C*d - 4*B*c*d^2 + 7*A
*d^3 - 4*c^3*D))/(6*b^3*d*(b*c - a*d)^3*(c + d*x)^(3/2)) - (A*b^3 - a*(b^2*B - a
*b*C + a^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*(c + d*x)^(3/2)) + (a^2*b*C*d^2 +
b^3*(2*c^2*C - 4*B*c*d + 7*A*d^2) - a^3*d^2*D + a*b^2*(4*c*C*d - 3*B*d^2 - 6*c^2
*D))/(b^2*(b*c - a*d)^4*Sqrt[c + d*x]) - ((b^3*(4*B*c - 7*A*d) - a*b^2*(8*c*C -
3*B*d) - 5*a^3*d*D + a^2*b*(C*d + 12*c*D))*Sqrt[c + d*x])/(4*b*(b*c - a*d)^4*(a
+ b*x)) - ((b^3*(8*c^2*C - 20*B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4
*c*D) + 3*a*b^2*(8*c*C*d - 5*B*d^2 - 8*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/S
qrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(9/2))

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

[Out]

Timed out

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Mathematica [A]  time = 2.35608, size = 360, normalized size = 0.82 \[ \sqrt{c+d x} \left (\frac{a \left (a^2 D-a b C+b^2 B\right )-A b^3}{2 b (a+b x)^2 (b c-a d)^3}+\frac{a^3 d D+3 a^2 b (C d-4 c D)+a b^2 (8 c C-7 B d)+b^3 (11 A d-4 B c)}{4 b (a+b x) (b c-a d)^4}+\frac{2 \left (b \left (3 A d^2-2 B c d+c^2 C\right )-a \left (B d^2+3 c^2 D-2 c C d\right )\right )}{(c+d x) (b c-a d)^4}+\frac{2 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{3 d (c+d x)^2 (a d-b c)^3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^3 d^2 D+3 a^2 b d (C d-4 c D)-3 a b^2 \left (5 B d^2+8 c^2 D-8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x]*((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))/(2*b*(b*c - a*d)^3*(a + b*
x)^2) + (b^3*(-4*B*c + 11*A*d) + a*b^2*(8*c*C - 7*B*d) + a^3*d*D + 3*a^2*b*(C*d
- 4*c*D))/(4*b*(b*c - a*d)^4*(a + b*x)) + (2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3
*D))/(3*d*(-(b*c) + a*d)^3*(c + d*x)^2) + (2*(b*(c^2*C - 2*B*c*d + 3*A*d^2) - a*
(-2*c*C*d + B*d^2 + 3*c^2*D)))/((b*c - a*d)^4*(c + d*x))) - ((b^3*(8*c^2*C - 20*
B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4*c*D) - 3*a*b^2*(-8*c*C*d + 5*
B*d^2 + 8*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(
b*c - a*d)^(9/2))

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Maple [B]  time = 0.046, size = 1376, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*C*a*b^2*c-d/(a*d-b*c)^4/(b*d*x+a*d)^
2*(d*x+c)^(3/2)*B*b^3*c+3/4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*a^2*b*C+
2/(a*d-b*c)^4*b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2)
)*C*c^2+11/4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*A*b^3+1/4*d^2/(a*d-b*c)
^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*a^3*D+5/4*d^3/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(
1/2)*C*a^3+35/4*d^2/(a*d-b*c)^4*b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/(
(a*d-b*c)*b)^(1/2))*A+3/4*d^2/(a*d-b*c)^4/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/
2)*b/((a*d-b*c)*b)^(1/2))*a^2*C+4*d/(a*d-b*c)^4/(d*x+c)^(1/2)*C*a*c-4*d/(a*d-b*c
)^4/(d*x+c)^(1/2)*B*b*c-2/3*d^2/(a*d-b*c)^3/(d*x+c)^(3/2)*A-2/3/(a*d-b*c)^3/(d*x
+c)^(3/2)*C*c^2+6*d/(a*d-b*c)^4*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-3*d/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*D*a^2*b*c+5/
4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*b^2*(d*x+c)^(1/2)*B*a*c+3/4*d^2/(a*d-b*c)^4/(b*d
*x+a*d)^2*b*(d*x+c)^(1/2)*C*a^2*c-2*d/(a*d-b*c)^4/(b*d*x+a*d)^2*b^2*(d*x+c)^(1/2
)*C*a*c^2+3*d/(a*d-b*c)^4/(b*d*x+a*d)^2*b*(d*x+c)^(1/2)*D*a^2*c^2-6/(a*d-b*c)^4/
(d*x+c)^(1/2)*D*a*c^2+2/3/d/(a*d-b*c)^3/(d*x+c)^(3/2)*D*c^3+2/3*d/(a*d-b*c)^3/(d
*x+c)^(3/2)*B*c+6*d^2/(a*d-b*c)^4/(d*x+c)^(1/2)*A*b-2*d^2/(a*d-b*c)^4/(d*x+c)^(1
/2)*B*a+2/(a*d-b*c)^4/(d*x+c)^(1/2)*C*b*c^2+13/4*d^3/(a*d-b*c)^4/(b*d*x+a*d)^2*b
^2*(d*x+c)^(1/2)*A*a-13/4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*b^3*(d*x+c)^(1/2)*A*c-9/
4*d^3/(a*d-b*c)^4/(b*d*x+a*d)^2*b*(d*x+c)^(1/2)*B*a^2+d/(a*d-b*c)^4/(b*d*x+a*d)^
2*b^3*(d*x+c)^(1/2)*B*c^2-1/4*d^3/(a*d-b*c)^4/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)*D*a^
4-11/4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(1/2)*D*a^3*c-15/4*d^2/(a*d-b*c)^4*
b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a-5*d/(a*d-b
*c)^4*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+1/
4*d^2/(a*d-b*c)^4/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-3*d/(a*d-b*c)^4/((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-7/4*d^2/(a*d-b*c)^4/(b*d*x+a*d)^2*(d*x+c)^(3/2)*B*a*b^2-6/(a*d
-b*c)^4*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*D*a*c^
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((D*x^3 + C*x^2 + B*x + A)/((b*x + a)^3*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268619, 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)^3*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(3*(8*(3*D*a^3*b^2 - C*a^2*b^3)*c^3*d + 4*(3*D*a^4*b - 6*C*a^3*b^2 + 5*B*
a^2*b^3)*c^2*d^2 - (D*a^5 + 3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^3)*c*d^3 + (8*
(3*D*a*b^4 - C*b^5)*c^2*d^2 + 4*(3*D*a^2*b^3 - 6*C*a*b^4 + 5*B*b^5)*c*d^3 - (D*a
^3*b^2 + 3*C*a^2*b^3 - 15*B*a*b^4 + 35*A*b^5)*d^4)*x^3 + (8*(3*D*a*b^4 - C*b^5)*
c^3*d + 20*(3*D*a^2*b^3 - 2*C*a*b^4 + B*b^5)*c^2*d^2 + (23*D*a^3*b^2 - 51*C*a^2*
b^3 + 55*B*a*b^4 - 35*A*b^5)*c*d^3 - 2*(D*a^4*b + 3*C*a^3*b^2 - 15*B*a^2*b^3 + 3
5*A*a*b^4)*d^4)*x^2 + (16*(3*D*a^2*b^3 - C*a*b^4)*c^3*d + 8*(6*D*a^3*b^2 - 7*C*a
^2*b^3 + 5*B*a*b^4)*c^2*d^2 + 10*(D*a^4*b - 3*C*a^3*b^2 + 5*B*a^2*b^3 - 7*A*a*b^
4)*c*d^3 - (D*a^5 + 3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*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*(8*D*a^2*b^2*c^4 + 8*A*a^3*b*d^4 + 2*(47*D*a^3*b - 25*C*a^2*
b^2 + 3*B*a*b^3 + 3*A*b^4)*c^3*d + (3*D*a^4 - 55*C*a^3*b + 83*B*a^2*b^2 - 39*A*a
*b^3)*c^2*d^2 + 16*(B*a^3*b - 5*A*a^2*b^2)*c*d^3 + 3*(8*(3*D*a*b^3 - C*b^4)*c^2*
d^2 + 4*(3*D*a^2*b^2 - 6*C*a*b^3 + 5*B*b^4)*c*d^3 - (D*a^3*b + 3*C*a^2*b^2 - 15*
B*a*b^3 + 35*A*b^4)*d^4)*x^3 + (8*D*b^4*c^4 + 32*(2*D*a*b^3 - C*b^4)*c^3*d + 8*(
27*D*a^2*b^2 - 17*C*a*b^3 + 10*B*b^4)*c^2*d^2 + 4*(6*D*a^3*b - 33*C*a^2*b^2 + 40
*B*a*b^3 - 35*A*b^4)*c*d^3 + (3*D*a^4 - 15*C*a^3*b + 75*B*a^2*b^2 - 175*A*a*b^3)
*d^4)*x^2 + (16*D*a*b^3*c^4 + 4*(41*D*a^2*b^2 - 22*C*a*b^3 + 3*B*b^4)*c^3*d + (1
29*D*a^3*b - 149*C*a^2*b^2 + 145*B*a*b^3 - 21*A*b^4)*c^2*d^2 + 2*(3*D*a^4 - 39*C
*a^3*b + 67*B*a^2*b^2 - 119*A*a*b^3)*c*d^3 + 8*(3*B*a^3*b - 7*A*a^2*b^2)*d^4)*x)
*sqrt(b^2*c - a*b*d))/((a^2*b^5*c^5*d - 4*a^3*b^4*c^4*d^2 + 6*a^4*b^3*c^3*d^3 -
4*a^5*b^2*c^2*d^4 + a^6*b*c*d^5 + (b^7*c^4*d^2 - 4*a*b^6*c^3*d^3 + 6*a^2*b^5*c^2
*d^4 - 4*a^3*b^4*c*d^5 + a^4*b^3*d^6)*x^3 + (b^7*c^5*d - 2*a*b^6*c^4*d^2 - 2*a^2
*b^5*c^3*d^3 + 8*a^3*b^4*c^2*d^4 - 7*a^4*b^3*c*d^5 + 2*a^5*b^2*d^6)*x^2 + (2*a*b
^6*c^5*d - 7*a^2*b^5*c^4*d^2 + 8*a^3*b^4*c^3*d^3 - 2*a^4*b^3*c^2*d^4 - 2*a^5*b^2
*c*d^5 + a^6*b*d^6)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c)), 1/12*(3*(8*(3*D*a^3*b
^2 - C*a^2*b^3)*c^3*d + 4*(3*D*a^4*b - 6*C*a^3*b^2 + 5*B*a^2*b^3)*c^2*d^2 - (D*a
^5 + 3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^3)*c*d^3 + (8*(3*D*a*b^4 - C*b^5)*c^2
*d^2 + 4*(3*D*a^2*b^3 - 6*C*a*b^4 + 5*B*b^5)*c*d^3 - (D*a^3*b^2 + 3*C*a^2*b^3 -
15*B*a*b^4 + 35*A*b^5)*d^4)*x^3 + (8*(3*D*a*b^4 - C*b^5)*c^3*d + 20*(3*D*a^2*b^3
 - 2*C*a*b^4 + B*b^5)*c^2*d^2 + (23*D*a^3*b^2 - 51*C*a^2*b^3 + 55*B*a*b^4 - 35*A
*b^5)*c*d^3 - 2*(D*a^4*b + 3*C*a^3*b^2 - 15*B*a^2*b^3 + 35*A*a*b^4)*d^4)*x^2 + (
16*(3*D*a^2*b^3 - C*a*b^4)*c^3*d + 8*(6*D*a^3*b^2 - 7*C*a^2*b^3 + 5*B*a*b^4)*c^2
*d^2 + 10*(D*a^4*b - 3*C*a^3*b^2 + 5*B*a^2*b^3 - 7*A*a*b^4)*c*d^3 - (D*a^5 + 3*C
*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*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))) - (8*D*a^2*b^2*c^4 + 8*A*a^3*b*d^4 + 2*(47
*D*a^3*b - 25*C*a^2*b^2 + 3*B*a*b^3 + 3*A*b^4)*c^3*d + (3*D*a^4 - 55*C*a^3*b + 8
3*B*a^2*b^2 - 39*A*a*b^3)*c^2*d^2 + 16*(B*a^3*b - 5*A*a^2*b^2)*c*d^3 + 3*(8*(3*D
*a*b^3 - C*b^4)*c^2*d^2 + 4*(3*D*a^2*b^2 - 6*C*a*b^3 + 5*B*b^4)*c*d^3 - (D*a^3*b
 + 3*C*a^2*b^2 - 15*B*a*b^3 + 35*A*b^4)*d^4)*x^3 + (8*D*b^4*c^4 + 32*(2*D*a*b^3
- C*b^4)*c^3*d + 8*(27*D*a^2*b^2 - 17*C*a*b^3 + 10*B*b^4)*c^2*d^2 + 4*(6*D*a^3*b
 - 33*C*a^2*b^2 + 40*B*a*b^3 - 35*A*b^4)*c*d^3 + (3*D*a^4 - 15*C*a^3*b + 75*B*a^
2*b^2 - 175*A*a*b^3)*d^4)*x^2 + (16*D*a*b^3*c^4 + 4*(41*D*a^2*b^2 - 22*C*a*b^3 +
 3*B*b^4)*c^3*d + (129*D*a^3*b - 149*C*a^2*b^2 + 145*B*a*b^3 - 21*A*b^4)*c^2*d^2
 + 2*(3*D*a^4 - 39*C*a^3*b + 67*B*a^2*b^2 - 119*A*a*b^3)*c*d^3 + 8*(3*B*a^3*b -
7*A*a^2*b^2)*d^4)*x)*sqrt(-b^2*c + a*b*d))/((a^2*b^5*c^5*d - 4*a^3*b^4*c^4*d^2 +
 6*a^4*b^3*c^3*d^3 - 4*a^5*b^2*c^2*d^4 + a^6*b*c*d^5 + (b^7*c^4*d^2 - 4*a*b^6*c^
3*d^3 + 6*a^2*b^5*c^2*d^4 - 4*a^3*b^4*c*d^5 + a^4*b^3*d^6)*x^3 + (b^7*c^5*d - 2*
a*b^6*c^4*d^2 - 2*a^2*b^5*c^3*d^3 + 8*a^3*b^4*c^2*d^4 - 7*a^4*b^3*c*d^5 + 2*a^5*
b^2*d^6)*x^2 + (2*a*b^6*c^5*d - 7*a^2*b^5*c^4*d^2 + 8*a^3*b^4*c^3*d^3 - 2*a^4*b^
3*c^2*d^4 - 2*a^5*b^2*c*d^5 + a^6*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)**3/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234238, size = 1035, normalized size = 2.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(24*D*a*b^2*c^2 - 8*C*b^3*c^2 + 12*D*a^2*b*c*d - 24*C*a*b^2*c*d + 20*B*b^3*
c*d - D*a^3*d^2 - 3*C*a^2*b*d^2 + 15*B*a*b^2*d^2 - 35*A*b^3*d^2)*arctan(sqrt(d*x
 + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*
a^3*b^2*c*d^3 + a^4*b*d^4)*sqrt(-b^2*c + a*b*d)) - 2/3*(D*b*c^4 + 9*(d*x + c)*D*
a*c^2*d - 3*(d*x + c)*C*b*c^2*d - D*a*c^3*d - C*b*c^3*d - 6*(d*x + c)*C*a*c*d^2
+ 6*(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 - 9*(d
*x + c)*A*b*d^3 - B*a*c*d^3 - A*b*c*d^3 + A*a*d^4)/((b^4*c^4*d - 4*a*b^3*c^3*d^2
 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*(d*x + c)^(3/2)) - 1/4*(12*(d*x
+ c)^(3/2)*D*a^2*b^2*c*d - 8*(d*x + c)^(3/2)*C*a*b^3*c*d + 4*(d*x + c)^(3/2)*B*b
^4*c*d - 12*sqrt(d*x + c)*D*a^2*b^2*c^2*d + 8*sqrt(d*x + c)*C*a*b^3*c^2*d - 4*sq
rt(d*x + c)*B*b^4*c^2*d - (d*x + c)^(3/2)*D*a^3*b*d^2 - 3*(d*x + c)^(3/2)*C*a^2*
b^2*d^2 + 7*(d*x + c)^(3/2)*B*a*b^3*d^2 - 11*(d*x + c)^(3/2)*A*b^4*d^2 + 11*sqrt
(d*x + c)*D*a^3*b*c*d^2 - 3*sqrt(d*x + c)*C*a^2*b^2*c*d^2 - 5*sqrt(d*x + c)*B*a*
b^3*c*d^2 + 13*sqrt(d*x + c)*A*b^4*c*d^2 + sqrt(d*x + c)*D*a^4*d^3 - 5*sqrt(d*x
+ c)*C*a^3*b*d^3 + 9*sqrt(d*x + c)*B*a^2*b^2*d^3 - 13*sqrt(d*x + c)*A*a*b^3*d^3)
/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*((
d*x + c)*b - b*c + a*d)^2)

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