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3.1177 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=314 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt{c} e^5}+\frac{3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^5 \sqrt{c d-b e}}-\frac{3 \sqrt{b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)}+\frac{\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2} \]

[Out]

(-3*(4*B*d*(2*c*d - b*e) - A*e*(4*c*d - b*e) + e*(4*B*c*d - b*B*e - 2*A*c*e)*x)*
Sqrt[b*x + c*x^2])/(4*e^4*(d + e*x)) + ((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(3/2
))/(2*e^2*(d + e*x)^2) - (3*(4*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 12*b*c*d*e
+ b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*Sqrt[c]*e^5) + (3*(A*e*(8
*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2))*Arc
Tanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*
Sqrt[d]*e^5*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.956924, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt{c} e^5}+\frac{3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^5 \sqrt{c d-b e}}-\frac{3 \sqrt{b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)}+\frac{\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(4*B*d*(2*c*d - b*e) - A*e*(4*c*d - b*e) + e*(4*B*c*d - b*B*e - 2*A*c*e)*x)*
Sqrt[b*x + c*x^2])/(4*e^4*(d + e*x)) + ((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(3/2
))/(2*e^2*(d + e*x)^2) - (3*(4*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 12*b*c*d*e
+ b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*Sqrt[c]*e^5) + (3*(A*e*(8
*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2))*Arc
Tanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*
Sqrt[d]*e^5*Sqrt[c*d - b*e])

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Rubi in Sympy [A]  time = 169.744, size = 328, normalized size = 1.04 \[ - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (2 A e - 4 B d - 2 B e x\right )}{4 e^{2} \left (d + e x\right )^{2}} - \frac{3 \sqrt{b x + c x^{2}} \left (2 A b e^{2} - 8 A c d e - 8 B b d e + 16 B c d^{2} - 2 e x \left (2 A c e + B b e - 4 B c d\right )\right )}{8 e^{4} \left (d + e x\right )} + \frac{3 \left (A b^{2} e^{3} - 8 A b c d e^{2} + 8 A c^{2} d^{2} e - 5 B b^{2} d e^{2} + 20 B b c d^{2} e - 16 B c^{2} d^{3}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 \sqrt{d} e^{5} \sqrt{b e - c d}} + \frac{3 \left (2 b c e \left (A e - 2 B d\right ) + \left (b e - 4 c d\right ) \left (2 A c e + B b e - 4 B c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 \sqrt{c} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**3,x)

[Out]

-(b*x + c*x**2)**(3/2)*(2*A*e - 4*B*d - 2*B*e*x)/(4*e**2*(d + e*x)**2) - 3*sqrt(
b*x + c*x**2)*(2*A*b*e**2 - 8*A*c*d*e - 8*B*b*d*e + 16*B*c*d**2 - 2*e*x*(2*A*c*e
 + B*b*e - 4*B*c*d))/(8*e**4*(d + e*x)) + 3*(A*b**2*e**3 - 8*A*b*c*d*e**2 + 8*A*
c**2*d**2*e - 5*B*b**2*d*e**2 + 20*B*b*c*d**2*e - 16*B*c**2*d**3)*atan((-b*d + x
*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(8*sqrt(d)*e**5*
sqrt(b*e - c*d)) + 3*(2*b*c*e*(A*e - 2*B*d) + (b*e - 4*c*d)*(2*A*c*e + B*b*e - 4
*B*c*d))*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*sqrt(c)*e**5)

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Mathematica [A]  time = 1.63114, size = 334, normalized size = 1.06 \[ \frac{(x (b+c x))^{3/2} \left (\frac{3 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (4 A c e (b e-2 c d)+B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{\sqrt{c} (b+c x)^{3/2}}-\frac{3 \left (B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} (b+c x)^{3/2} \sqrt{b e-c d}}+\frac{e \sqrt{x} \left (A e \left (2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )-b e (3 d+5 e x)\right )+B \left (b e \left (12 d^2+19 d e x+5 e^2 x^2\right )-2 c \left (12 d^3+18 d^2 e x+4 d e^2 x^2-e^3 x^3\right )\right )\right )}{(b+c x) (d+e x)^2}\right )}{4 e^5 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(A*e*(-(b*e*(3*d + 5*e*x)) + 2*c*(6*d^2 + 9*d*e
*x + 2*e^2*x^2)) + B*(b*e*(12*d^2 + 19*d*e*x + 5*e^2*x^2) - 2*c*(12*d^3 + 18*d^2
*e*x + 4*d*e^2*x^2 - e^3*x^3))))/((b + c*x)*(d + e*x)^2) - (3*(-(A*e*(8*c^2*d^2
- 8*b*c*d*e + b^2*e^2)) + B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2))*ArcTan[(Sqr
t[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) + b*e]*(
b + c*x)^(3/2)) + (3*(4*A*c*e*(-2*c*d + b*e) + B*(16*c^2*d^2 - 12*b*c*d*e + b^2*
e^2))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[c]*(b + c*x)^(3/2))))/(4*e^5
*x^(3/2))

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Maple [B]  time = 0.021, size = 7365, normalized size = 23.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^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)^(3/2)*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79145, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(2*B*c*e^4*x^3 - 24*B*c*d^3*e - 3*A*b*d*e^3 + 12*(B*b + A*c)*d^2*e^2 - (
8*B*c*d*e^3 - (5*B*b + 4*A*c)*e^4)*x^2 - (36*B*c*d^2*e^2 + 5*A*b*e^4 - (19*B*b +
 18*A*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)*sqrt(c) + 3*(16*B*c^2*d
^4 - 4*(3*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 4*A*b*c)*d^2*e^2 + (16*B*c^2*d^2*e^2
 - 4*(3*B*b*c + 2*A*c^2)*d*e^3 + (B*b^2 + 4*A*b*c)*e^4)*x^2 + 2*(16*B*c^2*d^3*e
- 4*(3*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 4*A*b*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e
)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 3*(16*B*c^2*d^5 - A*b^2*d^2
*e^3 - 4*(5*B*b*c + 2*A*c^2)*d^4*e + (5*B*b^2 + 8*A*b*c)*d^3*e^2 + (16*B*c^2*d^3
*e^2 - A*b^2*e^5 - 4*(5*B*b*c + 2*A*c^2)*d^2*e^3 + (5*B*b^2 + 8*A*b*c)*d*e^4)*x^
2 + 2*(16*B*c^2*d^4*e - A*b^2*d*e^4 - 4*(5*B*b*c + 2*A*c^2)*d^3*e^2 + (5*B*b^2 +
 8*A*b*c)*d^2*e^3)*x)*sqrt(c)*log(-(2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - sqrt(c
*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((e^7*x^2 + 2*d*e^6*x + d^2*e
^5)*sqrt(c*d^2 - b*d*e)*sqrt(c)), 1/8*(2*(2*B*c*e^4*x^3 - 24*B*c*d^3*e - 3*A*b*d
*e^3 + 12*(B*b + A*c)*d^2*e^2 - (8*B*c*d*e^3 - (5*B*b + 4*A*c)*e^4)*x^2 - (36*B*
c*d^2*e^2 + 5*A*b*e^4 - (19*B*b + 18*A*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*
x^2 + b*x)*sqrt(c) + 6*(16*B*c^2*d^5 - A*b^2*d^2*e^3 - 4*(5*B*b*c + 2*A*c^2)*d^4
*e + (5*B*b^2 + 8*A*b*c)*d^3*e^2 + (16*B*c^2*d^3*e^2 - A*b^2*e^5 - 4*(5*B*b*c +
2*A*c^2)*d^2*e^3 + (5*B*b^2 + 8*A*b*c)*d*e^4)*x^2 + 2*(16*B*c^2*d^4*e - A*b^2*d*
e^4 - 4*(5*B*b*c + 2*A*c^2)*d^3*e^2 + (5*B*b^2 + 8*A*b*c)*d^2*e^3)*x)*sqrt(c)*ar
ctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 3*(16*B*c^2*d^4
- 4*(3*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 4*A*b*c)*d^2*e^2 + (16*B*c^2*d^2*e^2 -
4*(3*B*b*c + 2*A*c^2)*d*e^3 + (B*b^2 + 4*A*b*c)*e^4)*x^2 + 2*(16*B*c^2*d^3*e - 4
*(3*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 4*A*b*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*
log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((e^7*x^2 + 2*d*e^6*x + d^2*e^
5)*sqrt(-c*d^2 + b*d*e)*sqrt(c)), 1/8*(2*(2*B*c*e^4*x^3 - 24*B*c*d^3*e - 3*A*b*d
*e^3 + 12*(B*b + A*c)*d^2*e^2 - (8*B*c*d*e^3 - (5*B*b + 4*A*c)*e^4)*x^2 - (36*B*
c*d^2*e^2 + 5*A*b*e^4 - (19*B*b + 18*A*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x
^2 + b*x)*sqrt(-c) + 6*(16*B*c^2*d^4 - 4*(3*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 4*
A*b*c)*d^2*e^2 + (16*B*c^2*d^2*e^2 - 4*(3*B*b*c + 2*A*c^2)*d*e^3 + (B*b^2 + 4*A*
b*c)*e^4)*x^2 + 2*(16*B*c^2*d^3*e - 4*(3*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 4*A
*b*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + 3
*(16*B*c^2*d^5 - A*b^2*d^2*e^3 - 4*(5*B*b*c + 2*A*c^2)*d^4*e + (5*B*b^2 + 8*A*b*
c)*d^3*e^2 + (16*B*c^2*d^3*e^2 - A*b^2*e^5 - 4*(5*B*b*c + 2*A*c^2)*d^2*e^3 + (5*
B*b^2 + 8*A*b*c)*d*e^4)*x^2 + 2*(16*B*c^2*d^4*e - A*b^2*d*e^4 - 4*(5*B*b*c + 2*A
*c^2)*d^3*e^2 + (5*B*b^2 + 8*A*b*c)*d^2*e^3)*x)*sqrt(-c)*log(-(2*(c*d^2 - b*d*e)
*sqrt(c*x^2 + b*x) - sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((
e^7*x^2 + 2*d*e^6*x + d^2*e^5)*sqrt(c*d^2 - b*d*e)*sqrt(-c)), 1/4*((2*B*c*e^4*x^
3 - 24*B*c*d^3*e - 3*A*b*d*e^3 + 12*(B*b + A*c)*d^2*e^2 - (8*B*c*d*e^3 - (5*B*b
+ 4*A*c)*e^4)*x^2 - (36*B*c*d^2*e^2 + 5*A*b*e^4 - (19*B*b + 18*A*c)*d*e^3)*x)*sq
rt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)*sqrt(-c) + 3*(16*B*c^2*d^5 - A*b^2*d^2*e^3
- 4*(5*B*b*c + 2*A*c^2)*d^4*e + (5*B*b^2 + 8*A*b*c)*d^3*e^2 + (16*B*c^2*d^3*e^2
- A*b^2*e^5 - 4*(5*B*b*c + 2*A*c^2)*d^2*e^3 + (5*B*b^2 + 8*A*b*c)*d*e^4)*x^2 + 2
*(16*B*c^2*d^4*e - A*b^2*d*e^4 - 4*(5*B*b*c + 2*A*c^2)*d^3*e^2 + (5*B*b^2 + 8*A*
b*c)*d^2*e^3)*x)*sqrt(-c)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d -
 b*e)*x)) + 3*(16*B*c^2*d^4 - 4*(3*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 4*A*b*c)*d^
2*e^2 + (16*B*c^2*d^2*e^2 - 4*(3*B*b*c + 2*A*c^2)*d*e^3 + (B*b^2 + 4*A*b*c)*e^4)
*x^2 + 2*(16*B*c^2*d^3*e - 4*(3*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 4*A*b*c)*d*e
^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((e^7*x^2
+ 2*d*e^6*x + d^2*e^5)*sqrt(-c*d^2 + b*d*e)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**3,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.660429, 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)^(3/2)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x

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