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

Optimal. Leaf size=279 \[ -\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^3 (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x} \left (-9 a^3 d D+a^2 b (12 c D+5 C d)-a b^2 (B d+8 c C)+b^3 (4 B c-3 A d)\right )}{4 b^3 (a+b x) (b c-a d)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-15 a^3 d^2 D+3 a^2 b d (12 c D+C d)-a b^2 \left (-B d^2+24 c^2 D+8 c C d\right )+b^3 \left (3 A d^2-4 B c d+8 c^2 C\right )\right )}{4 b^{7/2} (b c-a d)^{5/2}}+\frac{2 D \sqrt{c+d x}}{b^3 d} \]

[Out]

(2*D*Sqrt[c + d*x])/(b^3*d) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x]
)/(2*b^3*(b*c - a*d)*(a + b*x)^2) - ((b^3*(4*B*c - 3*A*d) - a*b^2*(8*c*C + B*d)
- 9*a^3*d*D + a^2*b*(5*C*d + 12*c*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)^2*(a + b
*x)) - ((b^3*(8*c^2*C - 4*B*c*d + 3*A*d^2) - 15*a^3*d^2*D + 3*a^2*b*d*(C*d + 12*
c*D) - a*b^2*(8*c*C*d - B*d^2 + 24*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(4*b^(7/2)*(b*c - a*d)^(5/2))

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Rubi [A]  time = 1.43404, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ -\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^3 (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x} \left (-9 a^3 d D+a^2 b (12 c D+5 C d)-a b^2 (B d+8 c C)+b^3 (4 B c-3 A d)\right )}{4 b^3 (a+b x) (b c-a d)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-15 a^3 d^2 D+3 a^2 b d (12 c D+C d)-a b^2 \left (-B d^2+24 c^2 D+8 c C d\right )+b^3 \left (3 A d^2-4 B c d+8 c^2 C\right )\right )}{4 b^{7/2} (b c-a d)^{5/2}}+\frac{2 D \sqrt{c+d x}}{b^3 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*D*Sqrt[c + d*x])/(b^3*d) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x]
)/(2*b^3*(b*c - a*d)*(a + b*x)^2) - ((b^3*(4*B*c - 3*A*d) - a*b^2*(8*c*C + B*d)
- 9*a^3*d*D + a^2*b*(5*C*d + 12*c*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)^2*(a + b
*x)) - ((b^3*(8*c^2*C - 4*B*c*d + 3*A*d^2) - 15*a^3*d^2*D + 3*a^2*b*d*(C*d + 12*
c*D) - a*b^2*(8*c*C*d - B*d^2 + 24*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(4*b^(7/2)*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 148.874, size = 347, normalized size = 1.24 \[ \frac{2 D \sqrt{c + d x}}{b^{3} d} + \frac{3 d \sqrt{c + d x} \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{4 b^{3} \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x} \left (B b^{2} - 2 C a b + 3 D a^{2}\right )}{b^{3} \left (a + b x\right ) \left (a d - b c\right )} + \frac{\sqrt{c + d x} \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{2 b^{3} \left (a + b x\right )^{2} \left (a d - b c\right )} + \frac{3 d^{2} \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{d \left (B b^{2} - 2 C a b + 3 D a^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{2 \left (C b - 3 D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}} \sqrt{a d - b c}} \]

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)**(1/2),x)

[Out]

2*D*sqrt(c + d*x)/(b**3*d) + 3*d*sqrt(c + d*x)*(A*b**3 - B*a*b**2 + C*a**2*b - D
*a**3)/(4*b**3*(a + b*x)*(a*d - b*c)**2) + sqrt(c + d*x)*(B*b**2 - 2*C*a*b + 3*D
*a**2)/(b**3*(a + b*x)*(a*d - b*c)) + sqrt(c + d*x)*(A*b**3 - B*a*b**2 + C*a**2*
b - D*a**3)/(2*b**3*(a + b*x)**2*(a*d - b*c)) + 3*d**2*(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))/(4*b**(7/2)*(a*d - b*
c)**(5/2)) + d*(B*b**2 - 2*C*a*b + 3*D*a**2)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d
- b*c))/(b**(7/2)*(a*d - b*c)**(3/2)) + 2*(C*b - 3*D*a)*atan(sqrt(b)*sqrt(c + d
*x)/sqrt(a*d - b*c))/(b**(7/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 1.61323, size = 253, normalized size = 0.91 \[ \frac{\sqrt{c+d x} \left (\frac{2 \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{(a+b x)^2 (b c-a d)}+\frac{9 a^3 d D-a^2 b (12 c D+5 C d)+a b^2 (B d+8 c C)+b^3 (3 A d-4 B c)}{(a+b x) (b c-a d)^2}+\frac{8 D}{d}\right )}{4 b^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-15 a^3 d^2 D+3 a^2 b d (12 c D+C d)+a b^2 \left (B d^2-24 c^2 D-8 c C d\right )+b^3 \left (3 A d^2-4 B c d+8 c^2 C\right )\right )}{4 b^{7/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*((8*D)/d + (2*(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D)))/((b*c - a*d
)*(a + b*x)^2) + (b^3*(-4*B*c + 3*A*d) + a*b^2*(8*c*C + B*d) + 9*a^3*d*D - a^2*b
*(5*C*d + 12*c*D))/((b*c - a*d)^2*(a + b*x))))/(4*b^3) - ((b^3*(8*c^2*C - 4*B*c*
d + 3*A*d^2) - 15*a^3*d^2*D + 3*a^2*b*d*(C*d + 12*c*D) + a*b^2*(-8*c*C*d + B*d^2
- 24*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(7/2)*(b*c
- a*d)^(5/2))

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Maple [B]  time = 0.033, size = 1207, normalized size = 4.3 \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

2*D*(d*x+c)^(1/2)/b^3/d+3/4*d^2*b/(b*d*x+a*d)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x
+c)^(3/2)*A+1/4*d^2/(b*d*x+a*d)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*B*a-
d*b/(b*d*x+a*d)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*B*c-5/4*d^2/b/(b*d*x
+a*d)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*C*a^2+2*d/(b*d*x+a*d)^2/(a^2*d
^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*C*a*c+9/4*d^2/b^2/(b*d*x+a*d)^2/(a^2*d^2-2*a
*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a^3*D-3*d/b/(b*d*x+a*d)^2/(a^2*d^2-2*a*b*c*d+b^2*c
^2)*(d*x+c)^(3/2)*D*a^2*c+5/4*d^2/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(1/2)*A-1/4*d^
2/b/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(1/2)*B*a-d/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^
(1/2)*B*c-3/4*d^2/b^2/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(1/2)*C*a^2+2*d/b/(b*d*x+a
*d)^2/(a*d-b*c)*(d*x+c)^(1/2)*C*a*c+7/4*d^2/b^3/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^
(1/2)*a^3*D-3*d/b^2/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(1/2)*D*a^2*c+3/4*d^2/(a^2*d
^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(
1/2))*A+1/4*d^2/b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)
^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a-d/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1
/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*c+3/4*d^2/b^2/(a^2*d^2-2*a*b*c
*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2*
C-2*d/b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/(
(a*d-b*c)*b)^(1/2))*C*a*c+2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arct
an((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*C*c^2-15/4*d^2/b^3/(a^2*d^2-2*a*b*c*d+b^
2*c^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+9*d
/b^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((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-6/b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/((a*d-b*c)*b)^(1/2)*arc
tan((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*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243156, 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*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[1/8*(2*(8*D*a^2*b^2*c^2 - 2*(13*D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b^4)*c*d +
(15*D*a^4 - 3*C*a^3*b - B*a^2*b^2 + 5*A*a*b^3)*d^2 + 8*(D*b^4*c^2 - 2*D*a*b^3*c*
d + D*a^2*b^2*d^2)*x^2 + (16*D*a*b^3*c^2 - 4*(11*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*
c*d + (25*D*a^3*b - 5*C*a^2*b^2 + B*a*b^3 + 3*A*b^4)*d^2)*x)*sqrt(b^2*c - a*b*d)
*sqrt(d*x + c) - (8*(3*D*a^3*b^2 - C*a^2*b^3)*c^2*d - 4*(9*D*a^4*b - 2*C*a^3*b^2
- B*a^2*b^3)*c*d^2 + (15*D*a^5 - 3*C*a^4*b - B*a^3*b^2 - 3*A*a^2*b^3)*d^3 + (8*
(3*D*a*b^4 - C*b^5)*c^2*d - 4*(9*D*a^2*b^3 - 2*C*a*b^4 - B*b^5)*c*d^2 + (15*D*a^
3*b^2 - 3*C*a^2*b^3 - B*a*b^4 - 3*A*b^5)*d^3)*x^2 + 2*(8*(3*D*a^2*b^3 - C*a*b^4)
*c^2*d - 4*(9*D*a^3*b^2 - 2*C*a^2*b^3 - B*a*b^4)*c*d^2 + (15*D*a^4*b - 3*C*a^3*b
^2 - B*a^2*b^3 - 3*A*a*b^4)*d^3)*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^2*b^5*c^2*d - 2*a^3*b^4*c*
d^2 + a^4*b^3*d^3 + (b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*x^2 + 2*(a*b^6*c^2
*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*x)*sqrt(b^2*c - a*b*d)), 1/4*((8*D*a^2*b^2*c
^2 - 2*(13*D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b^4)*c*d + (15*D*a^4 - 3*C*a^3*b
- B*a^2*b^2 + 5*A*a*b^3)*d^2 + 8*(D*b^4*c^2 - 2*D*a*b^3*c*d + D*a^2*b^2*d^2)*x^2
+ (16*D*a*b^3*c^2 - 4*(11*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c*d + (25*D*a^3*b - 5*
C*a^2*b^2 + B*a*b^3 + 3*A*b^4)*d^2)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) + (8*(
3*D*a^3*b^2 - C*a^2*b^3)*c^2*d - 4*(9*D*a^4*b - 2*C*a^3*b^2 - B*a^2*b^3)*c*d^2 +
(15*D*a^5 - 3*C*a^4*b - B*a^3*b^2 - 3*A*a^2*b^3)*d^3 + (8*(3*D*a*b^4 - C*b^5)*c
^2*d - 4*(9*D*a^2*b^3 - 2*C*a*b^4 - B*b^5)*c*d^2 + (15*D*a^3*b^2 - 3*C*a^2*b^3 -
B*a*b^4 - 3*A*b^5)*d^3)*x^2 + 2*(8*(3*D*a^2*b^3 - C*a*b^4)*c^2*d - 4*(9*D*a^3*b
^2 - 2*C*a^2*b^3 - B*a*b^4)*c*d^2 + (15*D*a^4*b - 3*C*a^3*b^2 - B*a^2*b^3 - 3*A*
a*b^4)*d^3)*x)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))))/((a^2*
b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3 + (b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5
*d^3)*x^2 + 2*(a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*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)**3/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22322, size = 714, normalized size = 2.56 \[ -\frac{{\left (24 \, D a b^{2} c^{2} - 8 \, C b^{3} c^{2} - 36 \, D a^{2} b c d + 8 \, C a b^{2} c d + 4 \, B b^{3} c d + 15 \, D a^{3} d^{2} - 3 \, C a^{2} b d^{2} - B a b^{2} d^{2} - 3 \, A b^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \sqrt{-b^{2} c + a b d}} - \frac{12 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{2} b^{2} c d - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} C a b^{3} c d + 4 \,{\left (d x + c\right )}^{\frac{3}{2}} B b^{4} c d - 12 \, \sqrt{d x + c} D a^{2} b^{2} c^{2} d + 8 \, \sqrt{d x + c} C a b^{3} c^{2} d - 4 \, \sqrt{d x + c} B b^{4} c^{2} d - 9 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{3} b d^{2} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C a^{2} b^{2} d^{2} -{\left (d x + c\right )}^{\frac{3}{2}} B a b^{3} d^{2} - 3 \,{\left (d x + c\right )}^{\frac{3}{2}} A b^{4} d^{2} + 19 \, \sqrt{d x + c} D a^{3} b c d^{2} - 11 \, \sqrt{d x + c} C a^{2} b^{2} c d^{2} + 3 \, \sqrt{d x + c} B a b^{3} c d^{2} + 5 \, \sqrt{d x + c} A b^{4} c d^{2} - 7 \, \sqrt{d x + c} D a^{4} d^{3} + 3 \, \sqrt{d x + c} C a^{3} b d^{3} + \sqrt{d x + c} B a^{2} b^{2} d^{3} - 5 \, \sqrt{d x + c} A a b^{3} d^{3}}{4 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} + \frac{2 \, \sqrt{d x + c} D}{b^{3} d} \]

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*sqrt(d*x + c)),x, algorithm="giac")

[Out]

-1/4*(24*D*a*b^2*c^2 - 8*C*b^3*c^2 - 36*D*a^2*b*c*d + 8*C*a*b^2*c*d + 4*B*b^3*c*
d + 15*D*a^3*d^2 - 3*C*a^2*b*d^2 - B*a*b^2*d^2 - 3*A*b^3*d^2)*arctan(sqrt(d*x +
c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*sqrt(-b^2*c +
a*b*d)) - 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*sqrt(d*x + c)*B*b^4*c^2*d - 9*(d*x + c)^(3/2)*D*a^3*b*d^2 +
5*(d*x + c)^(3/2)*C*a^2*b^2*d^2 - (d*x + c)^(3/2)*B*a*b^3*d^2 - 3*(d*x + c)^(3/
2)*A*b^4*d^2 + 19*sqrt(d*x + c)*D*a^3*b*c*d^2 - 11*sqrt(d*x + c)*C*a^2*b^2*c*d^2
+ 3*sqrt(d*x + c)*B*a*b^3*c*d^2 + 5*sqrt(d*x + c)*A*b^4*c*d^2 - 7*sqrt(d*x + c)
*D*a^4*d^3 + 3*sqrt(d*x + c)*C*a^3*b*d^3 + sqrt(d*x + c)*B*a^2*b^2*d^3 - 5*sqrt(
d*x + c)*A*a*b^3*d^3)/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*((d*x + c)*b - b*c
+ a*d)^2) + 2*sqrt(d*x + c)*D/(b^3*d)