3.317 \(\int \frac{x^3 \sqrt{a+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=211 \[ \frac{\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac{d^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^5}-\frac{\sqrt{a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^4}-\frac{7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac{\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \]

[Out]

-((8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(8*c*e^4) - (7*d*(a + c*x^2
)^(3/2))/(12*c*e^2) + ((d + e*x)*(a + c*x^2)^(3/2))/(4*c*e^2) + ((8*c^2*d^4 + 4*
a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(3/2)*e^5) + (
d^3*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^5

_______________________________________________________________________________________

Rubi [A]  time = 0.739254, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac{d^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^5}-\frac{\sqrt{a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^4}-\frac{7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac{\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

-((8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(8*c*e^4) - (7*d*(a + c*x^2
)^(3/2))/(12*c*e^2) + ((d + e*x)*(a + c*x^2)^(3/2))/(4*c*e^2) + ((8*c^2*d^4 + 4*
a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(3/2)*e^5) + (
d^3*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^5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.5576, size = 240, normalized size = 1.14 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e} + \frac{a x \sqrt{a + c x^{2}}}{8 c e} + \frac{a d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c} e^{3}} + \frac{\sqrt{c} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{5}} - \frac{d^{3} \sqrt{a + c x^{2}}}{e^{4}} + \frac{d^{3} \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{5}} + \frac{d^{2} x \sqrt{a + c x^{2}}}{2 e^{3}} + \frac{x^{3} \sqrt{a + c x^{2}}}{4 e} - \frac{d \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**2+a)**(1/2)/(e*x+d),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.482812, size = 219, normalized size = 1.04 \[ \frac{3 \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )-24 c^{3/2} d^3 \sqrt{a e^2+c d^2} \log (d+e x)+\sqrt{c} \left (24 c d^3 \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+e \sqrt{a+c x^2} \left (a e^2 (3 e x-8 d)+c \left (-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )\right )\right )}{24 c^{3/2} e^5} \]

Antiderivative was successfully verified.

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

[Out]

(-24*c^(3/2)*d^3*Sqrt[c*d^2 + a*e^2]*Log[d + e*x] + 3*(8*c^2*d^4 + 4*a*c*d^2*e^2
- a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + Sqrt[c]*(e*Sqrt[a + c*x^2]*(a*e
^2*(-8*d + 3*e*x) + c*(-24*d^3 + 12*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3)) + 24*c*d
^3*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]]))/
(24*c^(3/2)*e^5)

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 515, normalized size = 2.4 \[{\frac{{d}^{2}x}{2\,{e}^{3}}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{2}a}{2\,{e}^{3}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{x}{4\,ce} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{8\,ce}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}}{8\,e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{d}{3\,c{e}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}}{{e}^{4}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}+{\frac{{d}^{4}}{{e}^{5}}\sqrt{c}\ln \left ({1 \left ( -{\frac{cd}{e}}+c \left ( x+{\frac{d}{e}} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) }+{\frac{{d}^{3}a}{{e}^{4}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{5}c}{{e}^{6}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^2+a)^(1/2)/(e*x+d),x)

[Out]

1/2*d^2/e^3*x*(c*x^2+a)^(1/2)+1/2*d^2/e^3*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)
)+1/4/e*x*(c*x^2+a)^(3/2)/c-1/8/e*a/c*x*(c*x^2+a)^(1/2)-1/8/e*a^2/c^(3/2)*ln(x*c
^(1/2)+(c*x^2+a)^(1/2))-1/3*d*(c*x^2+a)^(3/2)/c/e^2-d^3/e^4*((x+d/e)^2*c-2*c*d/e
*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+d^4/e^5*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^3/e^4/((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^5/e^6/((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

_______________________________________________________________________________________

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^3/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 6.77092, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(24*sqrt(c*d^2 + a*e^2)*c^(3/2)*d^3*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*(6*c*e^4*x^3 - 8*c*d*e^3*x^2 - 24*c*d^3*e - 8
*a*d*e^3 + 3*(4*c*d^2*e^2 + a*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) - 3*(8*c^2*d^4 + 4
*a*c*d^2*e^2 - a^2*e^4)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/(c^(
3/2)*e^5), -1/48*(48*sqrt(-c*d^2 - a*e^2)*c^(3/2)*d^3*arctan((c*d*x - a*e)/(sqrt
(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) - 2*(6*c*e^4*x^3 - 8*c*d*e^3*x^2 - 24*c*d^3*e
- 8*a*d*e^3 + 3*(4*c*d^2*e^2 + a*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 3*(8*c^2*d^4
+ 4*a*c*d^2*e^2 - a^2*e^4)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/
(c^(3/2)*e^5), 1/24*(12*sqrt(c*d^2 + a*e^2)*sqrt(-c)*c*d^3*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)) + (6*c*e^4*x^3 - 8*c*d*e^3*x^2 -
24*c*d^3*e - 8*a*d*e^3 + 3*(4*c*d^2*e^2 + a*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) +
3*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqr
t(-c)*c*e^5), -1/24*(24*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*c*d^3*arctan((c*d*x - a*e)
/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) - (6*c*e^4*x^3 - 8*c*d*e^3*x^2 - 24*c*d
^3*e - 8*a*d*e^3 + 3*(4*c*d^2*e^2 + a*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) - 3*(8*c^
2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c
*e^5)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{a + c x^{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(c*x**2+a)**(1/2)/(e*x+d),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.30515, size = 271, normalized size = 1.28 \[ -\frac{2 \,{\left (c d^{5} + a d^{3} e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-5\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{24} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x + \frac{3 \,{\left (4 \, c^{2} d^{2} e^{12} + a c e^{14}\right )} e^{\left (-15\right )}}{c^{2}}\right )} x - \frac{8 \,{\left (3 \, c^{2} d^{3} e^{11} + a c d e^{13}\right )} e^{\left (-15\right )}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d^5 + a*d^3*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqr
t(-c*d^2 - a*e^2))*e^(-5)/sqrt(-c*d^2 - a*e^2) + 1/24*sqrt(c*x^2 + a)*((2*(3*x*e
^(-1) - 4*d*e^(-2))*x + 3*(4*c^2*d^2*e^12 + a*c*e^14)*e^(-15)/c^2)*x - 8*(3*c^2*
d^3*e^11 + a*c*d*e^13)*e^(-15)/c^2) - 1/8*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*
e^(-5)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)