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

Optimal. Leaf size=198 \[ \frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c} e^4}-\frac{3 \sqrt{d} \sqrt{c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^4}-\frac{3 \sqrt{b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left (b x+c x^2\right )^{3/2}}{e (d+e x)} \]

[Out]

(-3*(4*c*d - 3*b*e - 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3) - (b*x + c*x^2)^(3/2)/(
e*(d + e*x)) + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(4*Sqrt[c]*e^4) - (3*Sqrt[d]*Sqrt[c*d - b*e]*(2*c*d - b*e)*ArcTanh[(
b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^4)

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Rubi [A]  time = 0.614746, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c} e^4}-\frac{3 \sqrt{d} \sqrt{c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^4}-\frac{3 \sqrt{b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left (b x+c x^2\right )^{3/2}}{e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(4*c*d - 3*b*e - 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3) - (b*x + c*x^2)^(3/2)/(
e*(d + e*x)) + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(4*Sqrt[c]*e^4) - (3*Sqrt[d]*Sqrt[c*d - b*e]*(2*c*d - b*e)*ArcTanh[(
b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^4)

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Rubi in Sympy [A]  time = 74.4644, size = 184, normalized size = 0.93 \[ - \frac{3 \sqrt{d} \left (b e - 2 c d\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 e^{4}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )} + \frac{3 \sqrt{b x + c x^{2}} \left (3 b e - 4 c d + 2 c e x\right )}{4 e^{3}} + \frac{3 \left (b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 \sqrt{c} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(d)*(b*e - 2*c*d)*sqrt(b*e - c*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d
)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(2*e**4) - (b*x + c*x**2)**(3/2)/(e*(d +
e*x)) + 3*sqrt(b*x + c*x**2)*(3*b*e - 4*c*d + 2*c*e*x)/(4*e**3) + 3*(b**2*e**2 -
8*b*c*d*e + 8*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*sqrt(c)*e**4)

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Mathematica [A]  time = 0.615634, size = 224, normalized size = 1.13 \[ \frac{(x (b+c x))^{3/2} \left (\frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} (b+c x)^{3/2}}-\frac{12 \sqrt{d} \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{3/2} \sqrt{b e-c d}}+\frac{e \sqrt{x} \left (b e (9 d+5 e x)-2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )}{(b+c x) (d+e x)}\right )}{4 e^4 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(b*e*(9*d + 5*e*x) - 2*c*(6*d^2 + 3*d*e*x - e^2
*x^2)))/((b + c*x)*(d + e*x)) - (12*Sqrt[d]*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*Ar
cTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*
(b + c*x)^(3/2)) + (3*(8*c^2*d^2 - 8*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^4*x^(3/2))

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Maple [B]  time = 0.015, size = 1569, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d/(b*e-c*d)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)+
15/2/e^4*d^3/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*
d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2))/(d/e+x))*b*c^2-1/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+
x)-d*(b*e-c*d)/e^2)^(3/2)*b+1/e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(3/2)*c-27/8/e^2*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1
/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2+6/e^3
*d^2/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b-6/e^3*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e
^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c+9/4/
e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2+3/e^3*
d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2+3/2/
e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b*c-3/2/
e^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2-
21/4/e^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b
*c+3/8/e/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^3-3/e^4*d^3/(b*e-c*d)*ln((1/2*(
b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2))*c^(5/2)+3/2/e^2*d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*
d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3-3/e^5*d^4/(b*e-c*d)/(-d*(b*e-c*d
)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(
1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^3-c/d
/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x

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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)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315752, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(12*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*lo
g((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) -
2*(2*c*e^3*x^2 - 12*c*d^2*e + 9*b*d*e^2 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 +
b*x)*sqrt(c) - 3*(8*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*
e^2 + b^2*e^3)*x)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((e^5*x + d*
e^4)*sqrt(c)), -1/8*(24*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(-c*d^2 + b*
d*e)*sqrt(c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - 2*(2*c*e^3*x
^2 - 12*c*d^2*e + 9*b*d*e^2 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c)
- 3*(8*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3
)*x)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((e^5*x + d*e^4)*sqrt(c))
, -1/4*(6*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*l
og((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d))
- (2*c*e^3*x^2 - 12*c*d^2*e + 9*b*d*e^2 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 +
b*x)*sqrt(-c) - 3*(8*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*
e^2 + b^2*e^3)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((e^5*x + d*e^4)*sqr
t(-c)), -1/4*(12*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sq
rt(-c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - (2*c*e^3*x^2 - 12*
c*d^2*e + 9*b*d*e^2 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(8
*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*x)*ar
ctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((e^5*x + d*e^4)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [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)^(3/2)/(e*x + d)^2,x, algorithm="giac")

[Out]

Timed out