3.787 \(\int \frac{x^7 \sqrt{c+d x^4}}{a+b x^4} \, dx\)

Optimal. Leaf size=93 \[ -\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d} \]

[Out]

-(a*Sqrt[c + d*x^4])/(2*b^2) + (c + d*x^4)^(3/2)/(6*b*d) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4]
)/Sqrt[b*c - a*d]])/(2*b^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0824442, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 80, 50, 63, 208} \[ -\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

-(a*Sqrt[c + d*x^4])/(2*b^2) + (c + d*x^4)^(3/2)/(6*b*d) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4]
)/Sqrt[b*c - a*d]])/(2*b^(5/2))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x^7 \sqrt{c+d x^4}}{a+b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x \sqrt{c+d x}}{a+b x} \, dx,x,x^4\right )\\ &=\frac{\left (c+d x^4\right )^{3/2}}{6 b d}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{a+b x} \, dx,x,x^4\right )}{4 b}\\ &=-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d}-\frac{(a (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{4 b^2}\\ &=-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d}-\frac{(a (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^4}\right )}{2 b^2 d}\\ &=-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0650025, size = 88, normalized size = 0.95 \[ \frac{\sqrt{c+d x^4} \left (b \left (c+d x^4\right )-3 a d\right )}{6 b^2 d}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(Sqrt[c + d*x^4]*(-3*a*d + b*(c + d*x^4)))/(6*b^2*d) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sq
rt[b*c - a*d]])/(2*b^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.054, size = 1015, normalized size = 10.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.47033, size = 428, normalized size = 4.6 \begin{align*} \left [\frac{3 \, a d \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \,{\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt{d x^{4} + c}}{12 \, b^{2} d}, \frac{3 \, a d \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{4} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) +{\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt{d x^{4} + c}}{6 \, b^{2} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 20.6882, size = 90, normalized size = 0.97 \begin{align*} \frac{2 \left (- \frac{a d^{2} \sqrt{c + d x^{4}}}{4 b^{2}} + \frac{a d^{2} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{4 b^{3} \sqrt{\frac{a d - b c}{b}}} + \frac{d \left (c + d x^{4}\right )^{\frac{3}{2}}}{12 b}\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

2*(-a*d**2*sqrt(c + d*x**4)/(4*b**2) + a*d**2*(a*d - b*c)*atan(sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(4*b**3*s
qrt((a*d - b*c)/b)) + d*(c + d*x**4)**(3/2)/(12*b))/d**2

________________________________________________________________________________________

Giac [A]  time = 1.10728, size = 130, normalized size = 1.4 \begin{align*} -\frac{\frac{3 \,{\left (a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} - \frac{{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{2} - 3 \, \sqrt{d x^{4} + c} a b d}{b^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")

[Out]

-1/6*(3*(a*b*c*d - a^2*d^2)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) - ((d*x^
4 + c)^(3/2)*b^2 - 3*sqrt(d*x^4 + c)*a*b*d)/b^3)/d