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

Optimal. Leaf size=495 \[ -\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{96 b^3 (a+b x)^2 (b c-a d)^3}+\frac{\sqrt{c+d x} \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{24 b^3 (a+b x)^3 (b c-a d)^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{64 b^{7/2} (b c-a d)^{9/2}} \]

[Out]

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)*(a + b*x
)^4) - ((b^3*(8*B*c - 7*A*d) - a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*
d + 8*c*D))*Sqrt[c + d*x])/(24*b^3*(b*c - a*d)^2*(a + b*x)^3) - ((b^3*(48*c^2*C
- 40*B*c*d + 35*A*d^2) - 59*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C
*d - 5*B*d^2 + 144*c^2*D))*Sqrt[c + d*x])/(96*b^3*(b*c - a*d)^3*(a + b*x)^2) + (
(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*
D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*Sqrt[c + d*x])/(64*b^3
*(b*c - a*d)^4*(a + b*x)) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*
d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64
*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(7/2)*(b*c - a*
d)^(9/2))

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Rubi [A]  time = 2.15721, antiderivative size = 495, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{96 b^3 (a+b x)^2 (b c-a d)^3}+\frac{\sqrt{c+d x} \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{24 b^3 (a+b x)^3 (b c-a d)^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{64 b^{7/2} (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)*(a + b*x
)^4) - ((b^3*(8*B*c - 7*A*d) - a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*
d + 8*c*D))*Sqrt[c + d*x])/(24*b^3*(b*c - a*d)^2*(a + b*x)^3) - ((b^3*(48*c^2*C
- 40*B*c*d + 35*A*d^2) - 59*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C
*d - 5*B*d^2 + 144*c^2*D))*Sqrt[c + d*x])/(96*b^3*(b*c - a*d)^3*(a + b*x)^2) + (
(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*
D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*Sqrt[c + d*x])/(64*b^3
*(b*c - a*d)^4*(a + b*x)) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*
d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64
*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(7/2)*(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)**5/(d*x+c)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.26692, size = 449, normalized size = 0.91 \[ -\frac{\sqrt{c+d x} \left (48 (b c-a d)^3 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )+2 (a+b x)^2 (b c-a d) \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)+a b^2 \left (5 B d^2-144 c^2 D-16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )-3 (a+b x)^3 \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)+a b^2 d \left (5 B d^2+48 c^2 D-16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )+8 (a+b x) (b c-a d)^2 \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )\right )}{192 b^3 (a+b x)^4 (b c-a d)^4}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)+a b^2 d \left (5 B d^2+48 c^2 D-16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{64 b^{7/2} (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(48*(b*c - a*d)^3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D)) + 8*(b*c -
a*d)^2*(b^3*(8*B*c - 7*A*d) - a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*
d + 8*c*D))*(a + b*x) + 2*(b*c - a*d)*(b^3*(48*c^2*C - 40*B*c*d + 35*A*d^2) - 59
*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) + a*b^2*(-16*c*C*d + 5*B*d^2 - 144*c^2*D))
*(a + b*x)^2 - 3*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) + a*b^2*d*(-16*c*C*d +
5*B*d^2 + 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*(a +
b*x)^3))/(192*b^3*(b*c - a*d)^4*(a + b*x)^4) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C
*d - 8*c*D) + a*b^2*d*(-16*c*C*d + 5*B*d^2 + 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*
c*d^2 + 35*A*d^3 - 64*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/
(64*b^(7/2)*(b*c - a*d)^(9/2))

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Maple [B]  time = 0.039, size = 3252, normalized size = 6.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((D*x^3+C*x^2+B*x+A)/(b*x+a)^5/(d*x+c)^(1/2),x)

[Out]

11/8/(b*d*x+a*d)^4/b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*D*a^2*c*d^3+73/19
2/(b*d*x+a*d)^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a*B*d^4-11/8/(b*d*x+a*
d)^4/(a*d-b*c)*(d*x+c)^(1/2)*B*c*d^3+35/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+
6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*A*b^3*d^4+5/64/(b*d*x+a*d
)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2
)*a^3*d^4*D+385/192/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c
^3)*(d*x+c)^(5/2)*A*d^4+511/192/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x
+c)^(3/2)*A*d^4-d/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4
)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*D*c^3-5/8/(a^4
*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*
arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*c*d^3+3/4/(a^4*d^4-4*a^3*b*c*d^3+6
*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)
*b/((a*d-b*c)*b)^(1/2))*C*c^2*d^2-d/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*D*c^3+
5/4/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*C*c^2*d^2+11/64/(b*d*x+a*d)^4/(a^3*d^3
-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*a^2*C*d^4+3/4/(b*d*x+a*d)^4*
b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*D*a*c^2*d^2+3/4/(b
*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x
+c)^(7/2)*D*a*b^2*c^2*d^2-11/12/(b*d*x+a*d)^4*b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c
^2*d-b^3*c^3)*(d*x+c)^(5/2)*C*a*c*d^3-1/4/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6
*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*C*a*b^2*c*d^3-3/8/(b*d*x+a
*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7
/2)*D*a^2*b*c*d^3-3/8/b^2/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d
+b^4*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*d^3+3/4/b/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*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*c^2*d^2-1/4/b/(a
^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((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^3+1/4/(b*d*x+a*d)^4/(a*d-b
*c)/b*(d*x+c)^(1/2)*C*a*c*d^3+3/8/(b*d*x+a*d)^4/(a*d-b*c)/b^2*(d*x+c)^(1/2)*D*a^
2*c*d^3-3/4/(b*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c)^(1/2)*D*a*c^2*d^2+93/64/(b*d*x+a*d
)^4/(a*d-b*c)*(d*x+c)^(1/2)*A*d^4+35/64/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2
-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)
^(1/2))*A*d^4+3/64/b^2/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^
4*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*d^4
+5/64/b^3/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-
b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^3*d^4*D-5/8/(b*d*x+a
*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7
/2)*B*c*d^3*b^3+3/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*
b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*a^2*b*C*d^4-5/12/(b*d*x+a*d)^4/(a^2*d^2-2*a*b*c
*d+b^2*c^2)*(d*x+c)^(3/2)*C*a*c*d^3-3/4/(b*d*x+a*d)^4/(a^2*d^2-2*a*b*c*d+b^2*c^2
)*(d*x+c)^(3/2)*D*a*c^2*d^2-3/64/(b*d*x+a*d)^4/(a*d-b*c)/b^2*(d*x+c)^(1/2)*a^2*C
*d^4-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b^3*(d*x+c)^(1/2)*a^3*d^4*D+3/4/(b*d*x+a*d)^4/
(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*C*
c^2*d^2*b^3+55/192/(b*d*x+a*d)^4*b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)
*(d*x+c)^(5/2)*a*B*d^4+5/8/(b*d*x+a*d)^4/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^
3*c^3)*(d*x+c)^(5/2)*D*a^2*c*d^3-55/24/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+
3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*B*c*d^3+11/4/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a
^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*C*c^2*d^2-73/192/(b*d*x+a*d)^4/b
/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*a^3*d^4*D-73/24/(b*
d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*B*c*d^3-11/64/(b*d*x+a*d)
^4/b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a^2*C*d^4+13/4/(b*d*x+a*d)^4*b/(a
^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*C*c^2*d^2-55/192/(b*d*x+a*d)^4/b^2/(a^2*
d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a^3*d^4*D-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b*(d
*x+c)^(1/2)*a*B*d^4+5/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-
4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*a*b^2*B*d^4-3*d/(b*d*x+a*d)^4*b/(a^2*d^2-2*
a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*D*c^3-d/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*D*c^3*b^3-3*d/(b*d*x+a*d)^4*b
^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*D*c^3+5/64/b/(a^4
*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*
arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a*B*d^4

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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)^5*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260656, 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)^5*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[-1/384*(2*(16*(3*D*a^3*b^3 + C*a^2*b^4 + B*a*b^5 + 3*A*b^6)*c^3 + 8*(13*D*a^4*b
^2 - 11*C*a^3*b^3 - 9*B*a^2*b^4 - 25*A*a*b^5)*c^2*d - 2*(31*D*a^5*b + 21*C*a^4*b
^2 - 73*B*a^3*b^3 - 163*A*a^2*b^4)*c*d^2 + 3*(5*D*a^6 + 3*C*a^5*b + 5*B*a^4*b^2
- 93*A*a^3*b^3)*d^3 + 3*(64*D*b^6*c^3 - 48*(D*a*b^5 + C*b^6)*c^2*d + 8*(3*D*a^2*
b^4 + 2*C*a*b^5 + 5*B*b^6)*c*d^2 - (5*D*a^3*b^3 + 3*C*a^2*b^4 + 5*B*a*b^5 + 35*A
*b^6)*d^3)*x^3 + (96*(3*D*a*b^5 + C*b^6)*c^3 + 16*(12*D*a^2*b^4 - 35*C*a*b^5 - 5
*B*b^6)*c^2*d - 2*(119*D*a^3*b^3 - 91*C*a^2*b^4 - 225*B*a*b^5 - 35*A*b^6)*c*d^2
+ (73*D*a^4*b^2 - 33*C*a^3*b^3 - 55*B*a^2*b^4 - 385*A*a*b^5)*d^3)*x^2 + (64*(3*D
*a^2*b^4 + C*a*b^5 + B*b^6)*c^3 + 8*(37*D*a^3*b^3 - 45*C*a^2*b^4 - 37*B*a*b^5 -
7*A*b^6)*c^2*d - 4*(57*D*a^4*b^2 + 13*C*a^3*b^3 - 155*B*a^2*b^4 - 63*A*a*b^5)*c*
d^2 + (55*D*a^5*b + 33*C*a^4*b^2 - 73*B*a^3*b^3 - 511*A*a^2*b^4)*d^3)*x)*sqrt(b^
2*c - a*b*d)*sqrt(d*x + c) + 3*(64*D*a^4*b^3*c^3*d - 48*(D*a^5*b^2 + C*a^4*b^3)*
c^2*d^2 + 8*(3*D*a^6*b + 2*C*a^5*b^2 + 5*B*a^4*b^3)*c*d^3 - (5*D*a^7 + 3*C*a^6*b
+ 5*B*a^5*b^2 + 35*A*a^4*b^3)*d^4 + (64*D*b^7*c^3*d - 48*(D*a*b^6 + C*b^7)*c^2*
d^2 + 8*(3*D*a^2*b^5 + 2*C*a*b^6 + 5*B*b^7)*c*d^3 - (5*D*a^3*b^4 + 3*C*a^2*b^5 +
5*B*a*b^6 + 35*A*b^7)*d^4)*x^4 + 4*(64*D*a*b^6*c^3*d - 48*(D*a^2*b^5 + C*a*b^6)
*c^2*d^2 + 8*(3*D*a^3*b^4 + 2*C*a^2*b^5 + 5*B*a*b^6)*c*d^3 - (5*D*a^4*b^3 + 3*C*
a^3*b^4 + 5*B*a^2*b^5 + 35*A*a*b^6)*d^4)*x^3 + 6*(64*D*a^2*b^5*c^3*d - 48*(D*a^3
*b^4 + C*a^2*b^5)*c^2*d^2 + 8*(3*D*a^4*b^3 + 2*C*a^3*b^4 + 5*B*a^2*b^5)*c*d^3 -
(5*D*a^5*b^2 + 3*C*a^4*b^3 + 5*B*a^3*b^4 + 35*A*a^2*b^5)*d^4)*x^2 + 4*(64*D*a^3*
b^4*c^3*d - 48*(D*a^4*b^3 + C*a^3*b^4)*c^2*d^2 + 8*(3*D*a^5*b^2 + 2*C*a^4*b^3 +
5*B*a^3*b^4)*c*d^3 - (5*D*a^6*b + 3*C*a^5*b^2 + 5*B*a^4*b^3 + 35*A*a^3*b^4)*d^4)
*x)*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)))/((a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^
4*c*d^3 + a^8*b^3*d^4 + (b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b
^8*c*d^3 + a^4*b^7*d^4)*x^4 + 4*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^
2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*x^3 + 6*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^
4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*x^2 + 4*(a^3*b^8*c^4 - 4*a^4*b^7*
c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*x)*sqrt(b^2*c - a*b*d
)), -1/192*((16*(3*D*a^3*b^3 + C*a^2*b^4 + B*a*b^5 + 3*A*b^6)*c^3 + 8*(13*D*a^4*
b^2 - 11*C*a^3*b^3 - 9*B*a^2*b^4 - 25*A*a*b^5)*c^2*d - 2*(31*D*a^5*b + 21*C*a^4*
b^2 - 73*B*a^3*b^3 - 163*A*a^2*b^4)*c*d^2 + 3*(5*D*a^6 + 3*C*a^5*b + 5*B*a^4*b^2
- 93*A*a^3*b^3)*d^3 + 3*(64*D*b^6*c^3 - 48*(D*a*b^5 + C*b^6)*c^2*d + 8*(3*D*a^2
*b^4 + 2*C*a*b^5 + 5*B*b^6)*c*d^2 - (5*D*a^3*b^3 + 3*C*a^2*b^4 + 5*B*a*b^5 + 35*
A*b^6)*d^3)*x^3 + (96*(3*D*a*b^5 + C*b^6)*c^3 + 16*(12*D*a^2*b^4 - 35*C*a*b^5 -
5*B*b^6)*c^2*d - 2*(119*D*a^3*b^3 - 91*C*a^2*b^4 - 225*B*a*b^5 - 35*A*b^6)*c*d^2
+ (73*D*a^4*b^2 - 33*C*a^3*b^3 - 55*B*a^2*b^4 - 385*A*a*b^5)*d^3)*x^2 + (64*(3*
D*a^2*b^4 + C*a*b^5 + B*b^6)*c^3 + 8*(37*D*a^3*b^3 - 45*C*a^2*b^4 - 37*B*a*b^5 -
7*A*b^6)*c^2*d - 4*(57*D*a^4*b^2 + 13*C*a^3*b^3 - 155*B*a^2*b^4 - 63*A*a*b^5)*c
*d^2 + (55*D*a^5*b + 33*C*a^4*b^2 - 73*B*a^3*b^3 - 511*A*a^2*b^4)*d^3)*x)*sqrt(-
b^2*c + a*b*d)*sqrt(d*x + c) - 3*(64*D*a^4*b^3*c^3*d - 48*(D*a^5*b^2 + C*a^4*b^3
)*c^2*d^2 + 8*(3*D*a^6*b + 2*C*a^5*b^2 + 5*B*a^4*b^3)*c*d^3 - (5*D*a^7 + 3*C*a^6
*b + 5*B*a^5*b^2 + 35*A*a^4*b^3)*d^4 + (64*D*b^7*c^3*d - 48*(D*a*b^6 + C*b^7)*c^
2*d^2 + 8*(3*D*a^2*b^5 + 2*C*a*b^6 + 5*B*b^7)*c*d^3 - (5*D*a^3*b^4 + 3*C*a^2*b^5
+ 5*B*a*b^6 + 35*A*b^7)*d^4)*x^4 + 4*(64*D*a*b^6*c^3*d - 48*(D*a^2*b^5 + C*a*b^
6)*c^2*d^2 + 8*(3*D*a^3*b^4 + 2*C*a^2*b^5 + 5*B*a*b^6)*c*d^3 - (5*D*a^4*b^3 + 3*
C*a^3*b^4 + 5*B*a^2*b^5 + 35*A*a*b^6)*d^4)*x^3 + 6*(64*D*a^2*b^5*c^3*d - 48*(D*a
^3*b^4 + C*a^2*b^5)*c^2*d^2 + 8*(3*D*a^4*b^3 + 2*C*a^3*b^4 + 5*B*a^2*b^5)*c*d^3
- (5*D*a^5*b^2 + 3*C*a^4*b^3 + 5*B*a^3*b^4 + 35*A*a^2*b^5)*d^4)*x^2 + 4*(64*D*a^
3*b^4*c^3*d - 48*(D*a^4*b^3 + C*a^3*b^4)*c^2*d^2 + 8*(3*D*a^5*b^2 + 2*C*a^4*b^3
+ 5*B*a^3*b^4)*c*d^3 - (5*D*a^6*b + 3*C*a^5*b^2 + 5*B*a^4*b^3 + 35*A*a^3*b^4)*d^
4)*x)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))))/((a^4*b^7*c^4 -
4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4 + (b^11*c^4
- 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*x^4 + 4*(
a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4
)*x^3 + 6*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 +
a^6*b^5*d^4)*x^2 + 4*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6
*b^5*c*d^3 + a^7*b^4*d^4)*x)*sqrt(-b^2*c + a*b*d))]

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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)**5/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236352, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done