3.3.8 \(\int (d+e x) \sqrt {a+c x^4} \, dx\) [208]

Optimal. Leaf size=158 \[ \frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}+\frac {a^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}} \]

[Out]

1/4*a*e*arctanh(x^2*c^(1/2)/(c*x^4+a)^(1/2))/c^(1/2)+1/3*d*x*(c*x^4+a)^(1/2)+1/4*e*x^2*(c*x^4+a)^(1/2)+1/3*a^(
3/4)*d*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1
/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(1/4)/(c*x^4+a)
^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1899, 201, 226, 281, 223, 212} \begin {gather*} \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}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*Sqrt[a + c*x^4],x]

[Out]

(d*x*Sqrt[a + c*x^4])/3 + (e*x^2*Sqrt[a + c*x^4])/4 + (a*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(4*Sqrt[c])
+ (a^(3/4)*d*(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])/(3*c^(1/4)*Sqrt[a + c*x^4])

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 1899

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps

\begin {align*} \int (d+e x) \sqrt {a+c x^4} \, dx &=\int \left (d \sqrt {a+c x^4}+e x \sqrt {a+c x^4}\right ) \, dx\\ &=d \int \sqrt {a+c x^4} \, dx+e \int x \sqrt {a+c x^4} \, dx\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{3} (2 a d) \int \frac {1}{\sqrt {a+c x^4}} \, dx+\frac {1}{2} e \text {Subst}\left (\int \sqrt {a+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{4} (a e) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{4} (a e) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}+\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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.28, size = 109, normalized size = 0.69 \begin {gather*} \frac {\sqrt {a+c x^4} \left (\sqrt {c} e x^2 \sqrt {1+\frac {c x^4}{a}}+\sqrt {a} e \sinh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+4 \sqrt {c} d x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )\right )}{4 \sqrt {c} \sqrt {1+\frac {c x^4}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^4]*(Sqrt[c]*e*x^2*Sqrt[1 + (c*x^4)/a] + Sqrt[a]*e*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]] + 4*Sqrt[c]*d*x
*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^4)/a)]))/(4*Sqrt[c]*Sqrt[1 + (c*x^4)/a])

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 129, normalized size = 0.82

method	result	size
risch	\(\frac {x \left (3 e x +4 d \right ) \sqrt {c \,x^{4}+a}}{12}+\frac {a e \ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}+\frac {2 a d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(119\)
default	\(e \left (\frac {x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {a \ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\right )+d \left (\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\)	\(129\)
elliptic	\(\frac {e \,x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {d x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {a e \ln \left (2 x^{2} \sqrt {c}+2 \sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.12, size = 95, normalized size = 0.60 \begin {gather*} \frac {16 \, c^{\frac {3}{2}} d \left (-\frac {a}{c}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}, -1\right ) + 3 \, a \sqrt {c} e \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, \sqrt {c x^{4} + a} {\left (3 \, c x^{2} e + 4 \, c d x\right )}}{24 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/24*(16*c^(3/2)*d*(-a/c)^(3/4)*ellipticF((-a/c)^(1/4)/x, -1) + 3*a*sqrt(c)*e*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)
*sqrt(c)*x^2 - a) + 2*sqrt(c*x^4 + a)*(3*c*x^2*e + 4*c*d*x))/c

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 1.69, size = 88, normalized size = 0.56 \begin {gather*} \frac {\sqrt {a} d 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 {\sqrt {a} e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {a e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**4+a)**(1/2),x)

[Out]

sqrt(a)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(5/4)) + sqrt(a)*e*x**2*sq
rt(1 + c*x**4/a)/4 + a*e*asinh(sqrt(c)*x**2/sqrt(a))/(4*sqrt(c))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {c\,x^4+a}\,\left (d+e\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)*(d + e*x),x)

[Out]

int((a + c*x^4)^(1/2)*(d + e*x), x)

________________________________________________________________________________________