3.80 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{(a+b x)^3} \, dx\)

Optimal. Leaf size=658 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c \left (-A f^2+4 B e f+4 C e^2\right )+A d e f\right )\right )}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (c^2 \left (-\left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (6 a C d f-b (2 B d f+c C f+C d e))}{b^4 \sqrt{d} \sqrt{f}} \]

[Out]

-((12*a^3*C*d*f^2 - a^2*b*f*(17*C*d*e + 11*c*C*f + 4*B*d*f) + a*b^2*(B*f*(5*d*e
+ 3*c*f) + 4*C*e*(d*e + 4*c*f)) - b^3*(A*d*e*f + c*(4*C*e^2 + 4*B*e*f - A*f^2)))
*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^3*(b*c - a*d)*(b*e - a*f)^2) + ((6*a^3*C*d*f
- b^3*(4*B*c*e - A*d*e - A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + 3*B*c*f - 2*A*d*f)
- a^2*b*(2*B*d*f + 7*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*(b*c
- a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e +
f*x)^(3/2))/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - ((6*a*C*d*f - b*(C*d*e +
c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^
4*Sqrt[d]*Sqrt[f]) - ((24*a^4*C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + B*
f) + 2*c*d*e*(4*C*e + 3*B*f)) - 8*a^3*b*d*f*(B*d*f + 5*C*(d*e + c*f)) - b^4*(A*d
^2*e^2 - 2*c*d*e*(2*B*e + A*f) - c^2*(8*C*e^2 + 4*B*e*f - A*f^2)) + 3*a^2*b^2*(4
*B*d*f*(d*e + c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^2)))*ArcTanh[(Sqrt[b*e
- a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*b^4*(b*c - a*d)^(3/2)
*(b*e - a*f)^(3/2))

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Rubi [A]  time = 6.50657, antiderivative size = 657, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (c^2 \left (-\left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (6 a C d f-b (2 B d f+c C f+C d e))}{b^4 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

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

[Out]

-((12*a^3*C*d*f^2 - a^2*b*f*(17*C*d*e + 11*c*C*f + 4*B*d*f) - b^3*(4*c*C*e^2 + A
*d*e*f + c*f*(4*B*e - A*f)) + a*b^2*(B*f*(5*d*e + 3*c*f) + 4*C*e*(d*e + 4*c*f)))
*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^3*(b*c - a*d)*(b*e - a*f)^2) + ((6*a^3*C*d*f
- b^3*(4*B*c*e - A*d*e - A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + 3*B*c*f - 2*A*d*f)
- a^2*b*(2*B*d*f + 7*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*(b*c
- a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e +
f*x)^(3/2))/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - ((6*a*C*d*f - b*(C*d*e +
c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^
4*Sqrt[d]*Sqrt[f]) - ((24*a^4*C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + B*
f) + 2*c*d*e*(4*C*e + 3*B*f)) - 8*a^3*b*d*f*(B*d*f + 5*C*(d*e + c*f)) - b^4*(A*d
^2*e^2 - 2*c*d*e*(2*B*e + A*f) - c^2*(8*C*e^2 + 4*B*e*f - A*f^2)) + 3*a^2*b^2*(4
*B*d*f*(d*e + c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^2)))*ArcTanh[(Sqrt[b*e
- a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*b^4*(b*c - a*d)^(3/2)
*(b*e - a*f)^(3/2))

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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((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.85631, size = 1165, normalized size = 1.77 \[ \sqrt{c+d x} \sqrt{e+f x} \left (\frac{C}{b^3}+\frac{10 C d f a^3-9 b C d e a^2-9 b c C f a^2-6 b B d f a^2+8 b^2 c C e a+5 b^2 B d e a+5 b^2 B c f a+2 A b^2 d f a-4 b^3 B c e-A b^3 d e-A b^3 c f}{4 b^3 (b c-a d) (b e-a f) (a+b x)}+\frac{-C a^2+b B a-A b^2}{2 b^3 (a+b x)^2}\right )+\frac{\left (24 C d^2 f^2 a^4-8 b B d^2 f^2 a^3-40 b c C d f^2 a^3-40 b C d^2 e f a^3+15 b^2 C d^2 e^2 a^2+15 b^2 c^2 C f^2 a^2+12 b^2 B c d f^2 a^2+12 b^2 B d^2 e f a^2+66 b^2 c C d e f a^2-3 b^3 B d^2 e^2 a-24 b^3 c C d e^2 a-3 b^3 B c^2 f^2 a-24 b^3 c^2 C e f a-18 b^3 B c d e f a-A b^4 d^2 e^2+8 b^4 c^2 C e^2+4 b^4 B c d e^2-A b^4 c^2 f^2+4 b^4 B c^2 e f+2 A b^4 c d e f\right ) \log (a+b x)}{8 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}+\frac{\left (-\frac{3 C d^2 f^2 a^3}{b^4 (b c-a d) (b e-a f)}+\frac{B d^2 f^2 a^2}{b^3 (b c-a d) (b e-a f)}-\frac{d f (4 c C e+B d e+B c f) a}{b^2 (b c-a d) (b e-a f)}+\frac{C e f c^2+C d e^2 c}{2 b (b c-a d) (b e-a f)}+\frac{-a C d^2 e^2-a c^2 C f^2}{2 b^2 (b c-a d) (b e-a f)}+\frac{7 \left (a^2 C e f d^2+a^2 c C f^2 d\right )}{2 b^3 (b c-a d) (b e-a f)}+\frac{B c d e f}{b (b c-a d) (b e-a f)}\right ) \log \left (d e+c f+2 d f x+2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}\right )}{\sqrt{d} \sqrt{f}}-\frac{\left (24 C d^2 f^2 a^4-8 b B d^2 f^2 a^3-40 b c C d f^2 a^3-40 b C d^2 e f a^3+15 b^2 C d^2 e^2 a^2+15 b^2 c^2 C f^2 a^2+12 b^2 B c d f^2 a^2+12 b^2 B d^2 e f a^2+66 b^2 c C d e f a^2-3 b^3 B d^2 e^2 a-24 b^3 c C d e^2 a-3 b^3 B c^2 f^2 a-24 b^3 c^2 C e f a-18 b^3 B c d e f a-A b^4 d^2 e^2+8 b^4 c^2 C e^2+4 b^4 B c d e^2-A b^4 c^2 f^2+4 b^4 B c^2 e f+2 A b^4 c d e f\right ) \log \left (2 b c e-a d e+b d x e-a c f+b c f x-2 a d f x+2 \sqrt{b c-a d} \sqrt{b e-a f} \sqrt{c+d x} \sqrt{e+f x}\right )}{8 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x]*Sqrt[e + f*x]*(C/b^3 + (-(A*b^2) + a*b*B - a^2*C)/(2*b^3*(a + b*x)
^2) + (-4*b^3*B*c*e + 8*a*b^2*c*C*e - A*b^3*d*e + 5*a*b^2*B*d*e - 9*a^2*b*C*d*e
- A*b^3*c*f + 5*a*b^2*B*c*f - 9*a^2*b*c*C*f + 2*a*A*b^2*d*f - 6*a^2*b*B*d*f + 10
*a^3*C*d*f)/(4*b^3*(b*c - a*d)*(b*e - a*f)*(a + b*x))) + ((8*b^4*c^2*C*e^2 + 4*b
^4*B*c*d*e^2 - 24*a*b^3*c*C*d*e^2 - A*b^4*d^2*e^2 - 3*a*b^3*B*d^2*e^2 + 15*a^2*b
^2*C*d^2*e^2 + 4*b^4*B*c^2*e*f - 24*a*b^3*c^2*C*e*f + 2*A*b^4*c*d*e*f - 18*a*b^3
*B*c*d*e*f + 66*a^2*b^2*c*C*d*e*f + 12*a^2*b^2*B*d^2*e*f - 40*a^3*b*C*d^2*e*f -
A*b^4*c^2*f^2 - 3*a*b^3*B*c^2*f^2 + 15*a^2*b^2*c^2*C*f^2 + 12*a^2*b^2*B*c*d*f^2
- 40*a^3*b*c*C*d*f^2 - 8*a^3*b*B*d^2*f^2 + 24*a^4*C*d^2*f^2)*Log[a + b*x])/(8*b^
4*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2)) + (((B*c*d*e*f)/(b*(b*c - a*d)*(b*e - a*f
)) + (a^2*B*d^2*f^2)/(b^3*(b*c - a*d)*(b*e - a*f)) - (3*a^3*C*d^2*f^2)/(b^4*(b*c
- a*d)*(b*e - a*f)) - (a*d*f*(4*c*C*e + B*d*e + B*c*f))/(b^2*(b*c - a*d)*(b*e -
a*f)) + (c*C*d*e^2 + c^2*C*e*f)/(2*b*(b*c - a*d)*(b*e - a*f)) + (-(a*C*d^2*e^2)
- a*c^2*C*f^2)/(2*b^2*(b*c - a*d)*(b*e - a*f)) + (7*(a^2*C*d^2*e*f + a^2*c*C*d*
f^2))/(2*b^3*(b*c - a*d)*(b*e - a*f)))*Log[d*e + c*f + 2*d*f*x + 2*Sqrt[d]*Sqrt[
f]*Sqrt[c + d*x]*Sqrt[e + f*x]])/(Sqrt[d]*Sqrt[f]) - ((8*b^4*c^2*C*e^2 + 4*b^4*B
*c*d*e^2 - 24*a*b^3*c*C*d*e^2 - A*b^4*d^2*e^2 - 3*a*b^3*B*d^2*e^2 + 15*a^2*b^2*C
*d^2*e^2 + 4*b^4*B*c^2*e*f - 24*a*b^3*c^2*C*e*f + 2*A*b^4*c*d*e*f - 18*a*b^3*B*c
*d*e*f + 66*a^2*b^2*c*C*d*e*f + 12*a^2*b^2*B*d^2*e*f - 40*a^3*b*C*d^2*e*f - A*b^
4*c^2*f^2 - 3*a*b^3*B*c^2*f^2 + 15*a^2*b^2*c^2*C*f^2 + 12*a^2*b^2*B*c*d*f^2 - 40
*a^3*b*c*C*d*f^2 - 8*a^3*b*B*d^2*f^2 + 24*a^4*C*d^2*f^2)*Log[2*b*c*e - a*d*e - a
*c*f + b*d*e*x + b*c*f*x - 2*a*d*f*x + 2*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*Sqrt[c
+ d*x]*Sqrt[e + f*x]])/(8*b^4*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2))

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Maple [B]  time = 0.089, size = 12065, normalized size = 18.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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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((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.752322, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x