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

Optimal. Leaf size=216 \[ \frac{\sqrt{b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}-\frac{(2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac{d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^4}+\frac{\left (b x+c x^2\right )^{3/2}}{3 e} \]

[Out]

((8*c^2*d^2 - 10*b*c*d*e + b^2*e^2 - 2*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(
8*c*e^3) + (b*x + c*x^2)^(3/2)/(3*e) - ((2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - b
^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2)*e^4) + (d^(3/2)*(c*d
- b*e)^(3/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x
+ c*x^2])])/e^4

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Rubi [A]  time = 0.632135, antiderivative size = 216, 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 \[ \frac{\sqrt{b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}-\frac{(2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac{d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^4}+\frac{\left (b x+c x^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

((8*c^2*d^2 - 10*b*c*d*e + b^2*e^2 - 2*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(
8*c*e^3) + (b*x + c*x^2)^(3/2)/(3*e) - ((2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - b
^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2)*e^4) + (d^(3/2)*(c*d
- b*e)^(3/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x
+ c*x^2])])/e^4

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Rubi in Sympy [A]  time = 84.1416, size = 197, normalized size = 0.91 \[ \frac{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{4}} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 e} + \frac{\sqrt{b x + c x^{2}} \left (\frac{b^{2} e^{2}}{2} - 5 b c d e + 4 c^{2} d^{2} + c e x \left (b e - 2 c d\right )\right )}{4 c e^{3}} - \frac{\left (b e - 2 c d\right ) \left (b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.617855, size = 248, normalized size = 1.15 \[ \frac{\sqrt{x} \sqrt{b+c x} \left (\sqrt{b e-c d} \left (\sqrt{c} e \sqrt{x} \sqrt{b+c x} \left (3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-3 \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )+48 c^{3/2} d^{3/2} (c d-b e)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{24 c^{3/2} e^4 \sqrt{x (b+c x)} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(48*c^(3/2)*d^(3/2)*(c*d - b*e)^2*ArcTan[(Sqrt[-(c*d) + b
*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])] + Sqrt[-(c*d) + b*e]*(Sqrt[c]*e*Sqrt[x]*Sq
rt[b + c*x]*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^
2*x^2)) - 3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*Log[c*Sqrt[x
] + Sqrt[c]*Sqrt[b + c*x]])))/(24*c^(3/2)*e^4*Sqrt[-(c*d) + b*e]*Sqrt[x*(b + c*x
)])

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Maple [B]  time = 0.01, size = 1090, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)+1/4/e*(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-1/2/e^2*(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*d+1/8/e/c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2-5/4/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(1/2)*b*d-1/16/e/c^(3/2)*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))*b^3-3/8/e^2*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+1/e^3*d^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/e^3*d^2*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-1/e^4*d^3*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)-1/e^3*d^2/(-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+2/e^4*d^3/(-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-1/e^5*d^4/(-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^2

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.572991, 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)/(e*x + d),x, algorithm="fricas")

[Out]

[-1/48*(48*(c^2*d^2 - b*c*d*e)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log((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*(8*c^2*e^3*x^2 +
24*c^2*d^2*e - 30*b*c*d*e^2 + 3*b^2*e^3 - 2*(6*c^2*d*e^2 - 7*b*c*e^3)*x)*sqrt(c
*x^2 + b*x)*sqrt(c) - 3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*
log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/(c^(3/2)*e^4), 1/48*(96*(c^2*d
^2 - b*c*d*e)*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*(8*c^2*e^3*x^2 + 24*c^2*d^2*e - 30*b*c*d*e^2 + 3*b^2*e^3 - 2
*(6*c^2*d*e^2 - 7*b*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 3*(16*c^3*d^3 - 24*b*c
^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x
)*c))/(c^(3/2)*e^4), -1/24*(24*(c^2*d^2 - b*c*d*e)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*
log((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))
- (8*c^2*e^3*x^2 + 24*c^2*d^2*e - 30*b*c*d*e^2 + 3*b^2*e^3 - 2*(6*c^2*d*e^2 - 7
*b*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2
*c*d*e^2 + b^3*e^3)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c*e^4),
1/24*(48*(c^2*d^2 - b*c*d*e)*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)) + (8*c^2*e^3*x^2 + 24*c^2*d^2*e - 30*b*c*d*e^2 +
3*b^2*e^3 - 2*(6*c^2*d*e^2 - 7*b*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(16*c
^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*arctan(sqrt(c*x^2 + b*x)*sqrt
(-c)/(c*x)))/(sqrt(-c)*c*e^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError