3.318 \(\int \frac{x^2 \sqrt{a+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=153 \[ -\frac{d^2 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{d \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c} e^4}+\frac{d \sqrt{a+c x^2} (2 d-e x)}{2 e^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 c e} \]

[Out]

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

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Rubi [A]  time = 0.443296, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{d^2 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{d \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c} e^4}+\frac{d \sqrt{a+c x^2} (2 d-e x)}{2 e^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.1001, size = 167, normalized size = 1.09 \[ - \frac{a d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c} e^{2}} - \frac{\sqrt{c} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{4}} + \frac{d^{2} \sqrt{a + c x^{2}}}{e^{3}} - \frac{d^{2} \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4}} - \frac{d x \sqrt{a + c x^{2}}}{2 e^{2}} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**2+a)**(1/2)/(e*x+d),x)

[Out]

-a*d*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*sqrt(c)*e**2) - sqrt(c)*d**3*atanh(sqr
t(c)*x/sqrt(a + c*x**2))/e**4 + d**2*sqrt(a + c*x**2)/e**3 - d**2*sqrt(a*e**2 +
c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/e**4 - d*x
*sqrt(a + c*x**2)/(2*e**2) + (a + c*x**2)**(3/2)/(3*c*e)

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Mathematica [A]  time = 0.185174, size = 179, normalized size = 1.17 \[ \frac{e \sqrt{a+c x^2} \left (2 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 c d^2 \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-3 \sqrt{c} d \left (a e^2+2 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+6 c d^2 \sqrt{a e^2+c d^2} \log (d+e x)}{6 c e^4} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2]*(2*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 6*c*d^2*Sqrt[c*
d^2 + a*e^2]*Log[d + e*x] - 3*Sqrt[c]*d*(2*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt
[a + c*x^2]] - 6*c*d^2*Sqrt[c*d^2 + a*e^2]*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]
*Sqrt[a + c*x^2]])/(6*c*e^4)

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Maple [B]  time = 0.014, size = 448, normalized size = 2.9 \[{\frac{1}{3\,ce} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{dx}{2\,{e}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{ad}{2\,{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{d}^{2}}{{e}^{3}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}-{\frac{{d}^{3}}{{e}^{4}}\sqrt{c}\ln \left ({1 \left ( -{\frac{cd}{e}}+c \left ( x+{\frac{d}{e}} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) }-{\frac{{d}^{2}a}{{e}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{4}c}{{e}^{5}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2+a)^(1/2)/(e*x+d),x)

[Out]

1/3*(c*x^2+a)^(3/2)/c/e-1/2*d/e^2*x*(c*x^2+a)^(1/2)-1/2/e^2*d*a/c^(1/2)*ln(x*c^(
1/2)+(c*x^2+a)^(1/2))+d^2/e^3*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1
/2)-d^3/e^4*c^(1/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c-2*c*d/e*(x+d/e)+(
a*e^2+c*d^2)/e^2)^(1/2))-d^2/e^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e
^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e
^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a-d^4/e^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+
c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d
/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*c

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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(sqrt(c*x^2 + a)*x^2/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.571219, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \sqrt{c d^{2} + a e^{2}} c^{\frac{3}{2}} d^{2} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{12 \, c^{\frac{3}{2}} e^{4}}, \frac{12 \, \sqrt{-c d^{2} - a e^{2}} c^{\frac{3}{2}} d^{2} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) + 2 \,{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{12 \, c^{\frac{3}{2}} e^{4}}, \frac{3 \, \sqrt{c d^{2} + a e^{2}} \sqrt{-c} c d^{2} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) +{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c} - 3 \,{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{6 \, \sqrt{-c} c e^{4}}, \frac{6 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{-c} c d^{2} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) +{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c} - 3 \,{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{6 \, \sqrt{-c} c e^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(6*sqrt(c*d^2 + a*e^2)*c^(3/2)*d^2*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2
- (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a
))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(2*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e + 2*a*e
^3)*sqrt(c*x^2 + a)*sqrt(c) + 3*(2*c^2*d^3 + a*c*d*e^2)*log(2*sqrt(c*x^2 + a)*c*
x - (2*c*x^2 + a)*sqrt(c)))/(c^(3/2)*e^4), 1/12*(12*sqrt(-c*d^2 - a*e^2)*c^(3/2)
*d^2*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + 2*(2*c*e^3*x
^2 - 3*c*d*e^2*x + 6*c*d^2*e + 2*a*e^3)*sqrt(c*x^2 + a)*sqrt(c) + 3*(2*c^2*d^3 +
a*c*d*e^2)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/(c^(3/2)*e^4), 1
/6*(3*sqrt(c*d^2 + a*e^2)*sqrt(-c)*c*d^2*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2
- (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a
))/(e^2*x^2 + 2*d*e*x + d^2)) + (2*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e + 2*a*e^3
)*sqrt(c*x^2 + a)*sqrt(-c) - 3*(2*c^2*d^3 + a*c*d*e^2)*arctan(sqrt(-c)*x/sqrt(c*
x^2 + a)))/(sqrt(-c)*c*e^4), 1/6*(6*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*c*d^2*arctan((
c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + (2*c*e^3*x^2 - 3*c*d*e^2*
x + 6*c*d^2*e + 2*a*e^3)*sqrt(c*x^2 + a)*sqrt(-c) - 3*(2*c^2*d^3 + a*c*d*e^2)*ar
ctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c*e^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{a + c x^{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**2+a)**(1/2)/(e*x+d),x)

[Out]

Integral(x**2*sqrt(a + c*x**2)/(d + e*x), x)

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GIAC/XCAS [A]  time = 0.278268, size = 212, normalized size = 1.39 \[ \frac{2 \,{\left (c d^{4} + a d^{2} e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-4\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{{\left (2 \, c d^{3} + a d e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, x e^{\left (-1\right )} - 3 \, d e^{\left (-2\right )}\right )} x + \frac{2 \,{\left (3 \, c d^{2} e^{7} + a e^{9}\right )} e^{\left (-10\right )}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d^4 + a*d^2*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt
(-c*d^2 - a*e^2))*e^(-4)/sqrt(-c*d^2 - a*e^2) + 1/2*(2*c*d^3 + a*d*e^2)*e^(-4)*l
n(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/6*sqrt(c*x^2 + a)*((2*x*e^(-1)
- 3*d*e^(-2))*x + 2*(3*c*d^2*e^7 + a*e^9)*e^(-10)/c)