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

Optimal. Leaf size=188 \[ -\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2} \sqrt{b c-a d}}+\frac{2 \sqrt{c+d x} \left (a^2 d^2 D-a b d (C d-c D)+b^2 \left (-\left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3}+\frac{2 (c+d x)^{3/2} (-a d D-2 b c D+b C d)}{3 b^2 d^3}+\frac{2 D (c+d x)^{5/2}}{5 b d^3} \]

[Out]

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

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Rubi [A]  time = 0.401087, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ -\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2} \sqrt{b c-a d}}+\frac{2 \sqrt{c+d x} \left (a^2 d^2 D-a b d (C d-c D)+b^2 \left (-\left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3}+\frac{2 (c+d x)^{3/2} (-a d D-2 b c D+b C d)}{3 b^2 d^3}+\frac{2 D (c+d x)^{5/2}}{5 b d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 122.665, size = 187, normalized size = 0.99 \[ \frac{2 D \left (c + d x\right )^{\frac{5}{2}}}{5 b d^{3}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (C b d - D a d - 2 D b c\right )}{3 b^{2} d^{3}} + \frac{2 \sqrt{c + d x} \left (B b^{2} d^{2} - C a b d^{2} - C b^{2} c d + D a^{2} d^{2} + D a b c d + D b^{2} c^{2}\right )}{b^{3} d^{3}} + \frac{2 \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}} \sqrt{a d - b c}} \]

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

[Out]

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

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Mathematica [A]  time = 0.295404, size = 161, normalized size = 0.86 \[ \frac{2 \sqrt{c+d x} \left (15 a^2 d^2 D-5 a b d (-2 c D+3 C d+d D x)+b^2 \left (d^2 \left (15 B+5 C x+3 D x^2\right )+8 c^2 D-2 c d (5 C+2 D x)\right )\right )}{15 b^3 d^3}-\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 338, normalized size = 1.8 \[{\frac{2\,D}{5\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,C}{3\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Da}{3\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{4\,cD}{3\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{B\sqrt{dx+c}}{bd}}-2\,{\frac{Ca\sqrt{dx+c}}{{b}^{2}d}}-2\,{\frac{cC\sqrt{dx+c}}{b{d}^{2}}}+2\,{\frac{D{a}^{2}\sqrt{dx+c}}{d{b}^{3}}}+2\,{\frac{acD\sqrt{dx+c}}{{b}^{2}{d}^{2}}}+2\,{\frac{{c}^{2}D\sqrt{dx+c}}{b{d}^{3}}}+2\,{\frac{A}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{Ba}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{C{a}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{D{a}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228171, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} + 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right ) + 2 \,{\left (3 \, D b^{2} d^{2} x^{2} + 8 \, D b^{2} c^{2} + 10 \,{\left (D a b - C b^{2}\right )} c d + 15 \,{\left (D a^{2} - C a b + B b^{2}\right )} d^{2} -{\left (4 \, D b^{2} c d + 5 \,{\left (D a b - C b^{2}\right )} d^{2}\right )} x\right )} \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{15 \, \sqrt{b^{2} c - a b d} b^{3} d^{3}}, \frac{2 \,{\left (15 \,{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right ) +{\left (3 \, D b^{2} d^{2} x^{2} + 8 \, D b^{2} c^{2} + 10 \,{\left (D a b - C b^{2}\right )} c d + 15 \,{\left (D a^{2} - C a b + B b^{2}\right )} d^{2} -{\left (4 \, D b^{2} c d + 5 \,{\left (D a b - C b^{2}\right )} d^{2}\right )} x\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x + c}\right )}}{15 \, \sqrt{-b^{2} c + a b d} b^{3} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^3*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*(3*D*b^2*d^2*
x^2 + 8*D*b^2*c^2 + 10*(D*a*b - C*b^2)*c*d + 15*(D*a^2 - C*a*b + B*b^2)*d^2 - (4
*D*b^2*c*d + 5*(D*a*b - C*b^2)*d^2)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(sqrt(
b^2*c - a*b*d)*b^3*d^3), 2/15*(15*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^3*arctan
(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))) + (3*D*b^2*d^2*x^2 + 8*D*b^2
*c^2 + 10*(D*a*b - C*b^2)*c*d + 15*(D*a^2 - C*a*b + B*b^2)*d^2 - (4*D*b^2*c*d +
5*(D*a*b - C*b^2)*d^2)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))/(sqrt(-b^2*c + a*b
*d)*b^3*d^3)]

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Sympy [A]  time = 75.2973, size = 337, normalized size = 1.79 \[ \frac{2 D \left (c + d x\right )^{\frac{5}{2}}}{5 b d^{3}} - \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (- C b d + D a d + 2 D b c\right )}{3 b^{2} d^{3}} + \frac{2 \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{\frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{1}{c + d x} > - \frac{b}{a d - b c} \wedge \frac{b}{a d - b c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} < 0 \wedge \frac{1}{c + d x} < - \frac{b}{a d - b c} \end{cases}\right )}{b^{3}} + \frac{2 \sqrt{c + d x} \left (B b^{2} d^{2} - C a b d^{2} - C b^{2} c d + D a^{2} d^{2} + D a b c d + D b^{2} c^{2}\right )}{b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*D*(c + d*x)**(5/2)/(5*b*d**3) - 2*(c + d*x)**(3/2)*(-C*b*d + D*a*d + 2*D*b*c)/
(3*b**2*d**3) + 2*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*Piecewise((atan(1/(sq
rt(b/(a*d - b*c))*sqrt(c + d*x)))/(sqrt(b/(a*d - b*c))*(a*d - b*c)), b/(a*d - b*
c) > 0), (-acoth(1/(sqrt(-b/(a*d - b*c))*sqrt(c + d*x)))/(sqrt(-b/(a*d - b*c))*(
a*d - b*c)), (b/(a*d - b*c) < 0) & (1/(c + d*x) > -b/(a*d - b*c))), (-atanh(1/(s
qrt(-b/(a*d - b*c))*sqrt(c + d*x)))/(sqrt(-b/(a*d - b*c))*(a*d - b*c)), (b/(a*d
- b*c) < 0) & (1/(c + d*x) < -b/(a*d - b*c))))/b**3 + 2*sqrt(c + d*x)*(B*b**2*d*
*2 - C*a*b*d**2 - C*b**2*c*d + D*a**2*d**2 + D*a*b*c*d + D*b**2*c**2)/(b**3*d**3
)

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GIAC/XCAS [A]  time = 0.213326, size = 335, normalized size = 1.78 \[ -\frac{2 \,{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D b^{4} d^{12} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} D b^{4} c d^{12} + 15 \, \sqrt{d x + c} D b^{4} c^{2} d^{12} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} D a b^{3} d^{13} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C b^{4} d^{13} + 15 \, \sqrt{d x + c} D a b^{3} c d^{13} - 15 \, \sqrt{d x + c} C b^{4} c d^{13} + 15 \, \sqrt{d x + c} D a^{2} b^{2} d^{14} - 15 \, \sqrt{d x + c} C a b^{3} d^{14} + 15 \, \sqrt{d x + c} B b^{4} d^{14}\right )}}{15 \, b^{5} d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*
d))/(sqrt(-b^2*c + a*b*d)*b^3) + 2/15*(3*(d*x + c)^(5/2)*D*b^4*d^12 - 10*(d*x +
c)^(3/2)*D*b^4*c*d^12 + 15*sqrt(d*x + c)*D*b^4*c^2*d^12 - 5*(d*x + c)^(3/2)*D*a*
b^3*d^13 + 5*(d*x + c)^(3/2)*C*b^4*d^13 + 15*sqrt(d*x + c)*D*a*b^3*c*d^13 - 15*s
qrt(d*x + c)*C*b^4*c*d^13 + 15*sqrt(d*x + c)*D*a^2*b^2*d^14 - 15*sqrt(d*x + c)*C
*a*b^3*d^14 + 15*sqrt(d*x + c)*B*b^4*d^14)/(b^5*d^15)