3.183 \(\int (d+e x)^3 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=355 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (9 \sqrt{a} e^2+5 \sqrt{c} d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{6 a^{5/4} d e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{15} d x \sqrt{a+c x^4} \left (5 d^2+9 e^2 x^2\right )+\frac{3}{4} d^2 e x^2 \sqrt{a+c x^4}+\frac{3 a d^2 e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}}+\frac{6 a d e^2 x \sqrt{a+c x^4}}{5 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{e^3 \left (a+c x^4\right )^{3/2}}{6 c} \]

[Out]

(3*d^2*e*x^2*Sqrt[a + c*x^4])/4 + (6*a*d*e^2*x*Sqrt[a + c*x^4])/(5*Sqrt[c]*(Sqrt
[a] + Sqrt[c]*x^2)) + (d*x*(5*d^2 + 9*e^2*x^2)*Sqrt[a + c*x^4])/15 + (e^3*(a + c
*x^4)^(3/2))/(6*c) + (3*a*d^2*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(4*Sqrt[
c]) - (6*a^(5/4)*d*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[
c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*
x^4]) + (a^(3/4)*d*(5*Sqrt[c]*d^2 + 9*Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[
(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(15*c^(3/4)*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.503341, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (9 \sqrt{a} e^2+5 \sqrt{c} d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{6 a^{5/4} d e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{15} d x \sqrt{a+c x^4} \left (5 d^2+9 e^2 x^2\right )+\frac{3}{4} d^2 e x^2 \sqrt{a+c x^4}+\frac{3 a d^2 e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}}+\frac{6 a d e^2 x \sqrt{a+c x^4}}{5 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{e^3 \left (a+c x^4\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*Sqrt[a + c*x^4],x]

[Out]

(3*d^2*e*x^2*Sqrt[a + c*x^4])/4 + (6*a*d*e^2*x*Sqrt[a + c*x^4])/(5*Sqrt[c]*(Sqrt
[a] + Sqrt[c]*x^2)) + (d*x*(5*d^2 + 9*e^2*x^2)*Sqrt[a + c*x^4])/15 + (e^3*(a + c
*x^4)^(3/2))/(6*c) + (3*a*d^2*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(4*Sqrt[
c]) - (6*a^(5/4)*d*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[
c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*
x^4]) + (a^(3/4)*d*(5*Sqrt[c]*d^2 + 9*Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[
(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(15*c^(3/4)*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.5314, size = 332, normalized size = 0.94 \[ - \frac{6 a^{\frac{5}{4}} d e^{2} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{a^{\frac{3}{4}} d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (9 \sqrt{a} e^{2} + 5 \sqrt{c} d^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{3 a d^{2} e \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4 \sqrt{c}} + \frac{6 a d e^{2} x \sqrt{a + c x^{4}}}{5 \sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{3 d^{2} e x^{2} \sqrt{a + c x^{4}}}{4} + \frac{d x \sqrt{a + c x^{4}} \left (5 d^{2} + 9 e^{2} x^{2}\right )}{15} + \frac{e^{3} \left (a + c x^{4}\right )^{\frac{3}{2}}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**4+a)**(1/2),x)

[Out]

-6*a**(5/4)*d*e**2*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqr
t(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(5*c**(3/4)*sqrt(a + c*x
**4)) + a**(3/4)*d*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqr
t(c)*x**2)*(9*sqrt(a)*e**2 + 5*sqrt(c)*d**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/
4)), 1/2)/(15*c**(3/4)*sqrt(a + c*x**4)) + 3*a*d**2*e*atanh(sqrt(c)*x**2/sqrt(a
+ c*x**4))/(4*sqrt(c)) + 6*a*d*e**2*x*sqrt(a + c*x**4)/(5*sqrt(c)*(sqrt(a) + sqr
t(c)*x**2)) + 3*d**2*e*x**2*sqrt(a + c*x**4)/4 + d*x*sqrt(a + c*x**4)*(5*d**2 +
9*e**2*x**2)/15 + e**3*(a + c*x**4)**(3/2)/(6*c)

_______________________________________________________________________________________

Mathematica [C]  time = 0.724068, size = 310, normalized size = 0.87 \[ \frac{72 a^{3/2} \sqrt{c} d e^2 \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (10 a^2 e^3+45 a \sqrt{c} d^2 e \sqrt{a+c x^4} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )+a c x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+20 e^3 x^3\right )+c^2 x^5 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )-8 a \sqrt{c} d \sqrt{\frac{c x^4}{a}+1} \left (9 \sqrt{a} e^2+5 i \sqrt{c} d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{60 c \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[(I*Sqrt[c])/Sqrt[a]]*(10*a^2*e^3 + c^2*x^5*(20*d^3 + 45*d^2*e*x + 36*d*e^2
*x^2 + 10*e^3*x^3) + a*c*x*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 20*e^3*x^3) + 4
5*a*Sqrt[c]*d^2*e*Sqrt[a + c*x^4]*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]]) + 72*a
^(3/2)*Sqrt[c]*d*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sq
rt[a]]*x], -1] - 8*a*Sqrt[c]*d*((5*I)*Sqrt[c]*d^2 + 9*Sqrt[a]*e^2)*Sqrt[1 + (c*x
^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(60*Sqrt[(I*Sqrt[c
])/Sqrt[a]]*c*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.048, size = 334, normalized size = 0.9 \[{\frac{{d}^{3}x}{3}\sqrt{c{x}^{4}+a}}+{\frac{2\,a{d}^{3}}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{{e}^{3}}{6\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}e{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{3\,a{d}^{2}e}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{3\,d{e}^{2}{x}^{3}}{5}\sqrt{c{x}^{4}+a}}+{{\frac{6\,i}{5}}{e}^{2}d{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}}-{{\frac{6\,i}{5}}{e}^{2}d{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^4+a)^(1/2),x)

[Out]

1/3*d^3*x*(c*x^4+a)^(1/2)+2/3*d^3*a/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/
2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(
1/2)*c^(1/2))^(1/2),I)+1/6*e^3*(c*x^4+a)^(3/2)/c+3/4*d^2*e*x^2*(c*x^4+a)^(1/2)+3
/4*d^2*e*a/c^(1/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))+3/5*e^2*d*x^3*(c*x^4+a)^(1/2)
+6/5*I*e^2*d*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(
1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*EllipticF(x*(I/a^(1/2)*c^
(1/2))^(1/2),I)-6/5*I*e^2*d*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/
2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*EllipticE(
x*(I/a^(1/2)*c^(1/2))^(1/2),I)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + a}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + a)*(e*x + d)^3, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{4} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*(e*x + d)^3,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(c*x^4 + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 10.1082, size = 175, normalized size = 0.49 \[ \frac{\sqrt{a} d^{3} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{3 \sqrt{a} d^{2} e x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{3 \sqrt{a} d e^{2} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{3 a d^{2} e \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} + e^{3} \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: c = 0 \\\frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{6 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**4+a)**(1/2),x)

[Out]

sqrt(a)*d**3*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/a)/(
4*gamma(5/4)) + 3*sqrt(a)*d**2*e*x**2*sqrt(1 + c*x**4/a)/4 + 3*sqrt(a)*d*e**2*x*
*3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)
) + 3*a*d**2*e*asinh(sqrt(c)*x**2/sqrt(a))/(4*sqrt(c)) + e**3*Piecewise((sqrt(a)
*x**4/4, Eq(c, 0)), ((a + c*x**4)**(3/2)/(6*c), True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + a}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*(e*x + d)^3,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + a)*(e*x + d)^3, x)