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

Optimal. Leaf size=201 \[ -\frac{\sqrt{c+d x} \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{b^{7/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x} (-2 a d D-b c D+b C d)}{b^3 d^2}+\frac{2 D (c+d x)^{3/2}}{3 b^2 d^2} \]

[Out]

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

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Rubi [A]  time = 1.07579, antiderivative size = 201, 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 \[ -\frac{\sqrt{c+d x} \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{b^{7/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x} (-2 a d D-b c D+b C d)}{b^3 d^2}+\frac{2 D (c+d x)^{3/2}}{3 b^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 124.728, size = 228, normalized size = 1.13 \[ \frac{2 D \left (c + d x\right )^{\frac{3}{2}}}{3 b^{2} d^{2}} + \frac{\sqrt{c + d x} \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{b^{3} \left (a + b x\right ) \left (a d - b c\right )} + \frac{2 \sqrt{c + d x} \left (C b d - 2 D a d - D b c\right )}{b^{3} d^{2}} + \frac{d \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{2 \left (B b^{2} - 2 C a b + 3 D a^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}} \sqrt{a d - b c}} \]

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

[Out]

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

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Mathematica [A]  time = 0.924342, size = 186, normalized size = 0.93 \[ \frac{\sqrt{c+d x} \left (\frac{3 \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{(a+b x) (b c-a d)}+\frac{-12 a d D-4 b c D+6 b C d}{d^2}+\frac{2 b D x}{d}\right )}{3 b^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{b^{7/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 566, normalized size = 2.8 \[{\frac{2\,D}{3\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{C\sqrt{dx+c}}{{b}^{2}d}}-4\,{\frac{Da\sqrt{dx+c}}{{b}^{3}d}}-2\,{\frac{cD\sqrt{dx+c}}{{b}^{2}{d}^{2}}}+{\frac{Ad}{ \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{Bda}{ \left ( ad-bc \right ) b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{Cd{a}^{2}}{{b}^{2} \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{{a}^{3}dD}{{b}^{3} \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{Ad}{ad-bc}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{Bda}{ \left ( ad-bc \right ) b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{Bc}{ \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-3\,{\frac{Cd{a}^{2}}{{b}^{2} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{Cac}{ \left ( ad-bc \right ) b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+5\,{\frac{{a}^{3}dD}{{b}^{3} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{D{a}^{2}c}{{b}^{2} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

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)^(1/2),x)

[Out]

2/3*D*(d*x+c)^(3/2)/b^2/d^2+2/d/b^2*C*(d*x+c)^(1/2)-4/d/b^3*D*a*(d*x+c)^(1/2)-2/
d^2/b^2*D*c*(d*x+c)^(1/2)+d/(a*d-b*c)*(d*x+c)^(1/2)/(b*d*x+a*d)*A-d/b/(a*d-b*c)*
(d*x+c)^(1/2)/(b*d*x+a*d)*B*a+d/b^2/(a*d-b*c)*(d*x+c)^(1/2)/(b*d*x+a*d)*C*a^2-d/
b^3/(a*d-b*c)*(d*x+c)^(1/2)/(b*d*x+a*d)*D*a^3+d/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*ar
ctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*A+d/b/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*ar
ctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a-2/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*ar
ctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*c-3*d/b^2/(a*d-b*c)/((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/b/(a*d-b*c)/((a*d-b*c)*b
)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*C*a*c+5*d/b^3/(a*d-b*c)/((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/b^2/(a*d-b*c
)/((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*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250427, 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*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213616, size = 366, normalized size = 1.82 \[ \frac{{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - 5 \, D a^{3} d + 3 \, C a^{2} b d - B a b^{2} d - A b^{3} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{4} c - a b^{3} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x + c} D a^{3} d - \sqrt{d x + c} C a^{2} b d + \sqrt{d x + c} B a b^{2} d - \sqrt{d x + c} A b^{3} d}{{\left (b^{4} c - a b^{3} d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} D b^{4} d^{4} - 3 \, \sqrt{d x + c} D b^{4} c d^{4} - 6 \, \sqrt{d x + c} D a b^{3} d^{5} + 3 \, \sqrt{d x + c} C b^{4} d^{5}\right )}}{3 \, b^{6} d^{6}} \]

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*sqrt(d*x + c)),x, algorithm="giac")

[Out]

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