\(\int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2653]

Optimal result

Integrand size = 34, antiderivative size = 236 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+\frac {2 d \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

[Out]

1/72*b^8*c/a^4/(a*x+(a^2*x^2+b^2)^(1/2))^(9/2)-1/20*b^6*c/a^4/(a*x+(a^2*x^2+b^2)^(1/2))^(5/2)+2*d/(a*x+(a^2*x^
2+b^2)^(1/2))^(1/2)-1/12*b^2*c*(a*x+(a^2*x^2+b^2)^(1/2))^(3/2)/a^4+1/56*c*(a*x+(a^2*x^2+b^2)^(1/2))^(7/2)/a^4+
2*d*arctan((a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/b^(1/2))/b^(1/2)-2*d*arctanh((a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/b^(1/2
))/b^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6874, 2144, 464, 335, 304, 209, 212, 459} \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 d \arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {2 d}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\frac {b^2 c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{12 a^4}+\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}{56 a^4}+\frac {b^8 c}{72 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \]

[In]

Int[(d + c*x^4)/(x*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

(b^8*c)/(72*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)) - (b^6*c)/(20*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*d
)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - (b^2*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(12*a^4) + (c*(a*x + Sqrt[b^2 +
a^2*x^2])^(7/2))/(56*a^4) + (2*d*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b] - (2*d*ArcTanh[Sqrt[
a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b]

Rule 209

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

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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

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

Rule 459

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

Rule 464

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

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c x^3}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx \\ & = c \int \frac {x^3}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+d \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \\ & = \frac {c \text {Subst}\left (\int \frac {\left (-b^2+x^2\right )^3 \left (b^2+x^2\right )}{x^{11/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{16 a^4}+d \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2+x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = \frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \text {Subst}\left (\int \left (-\frac {b^8}{x^{11/2}}+\frac {2 b^6}{x^{7/2}}-2 b^2 \sqrt {x}+x^{5/2}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{16 a^4}+(2 d) \text {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = \frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+(4 d) \text {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right ) \\ & = \frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}-(2 d) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+(2 d) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right ) \\ & = \frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+\frac {2 d \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+\frac {2 d \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

[In]

Integrate[(d + c*x^4)/(x*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

(b^8*c)/(72*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)) - (b^6*c)/(20*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*d
)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - (b^2*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(12*a^4) + (c*(a*x + Sqrt[b^2 +
a^2*x^2])^(7/2))/(56*a^4) + (2*d*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b] - (2*d*ArcTanh[Sqrt[
a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b]

Maple [F]

\[\int \frac {c \,x^{4}+d}{x \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.04 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\left [\frac {630 \, a^{4} b^{\frac {3}{2}} d \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + 315 \, a^{4} b^{\frac {3}{2}} d \log \left (\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, {\left (35 \, a^{5} c x^{5} + a^{3} b^{2} c x^{3} - {\left (8 \, a b^{4} c - 315 \, a^{5} d\right )} x - {\left (35 \, a^{4} c x^{4} + 6 \, a^{2} b^{2} c x^{2} - 16 \, b^{4} c + 315 \, a^{4} d\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{315 \, a^{4} b^{2}}, \frac {630 \, a^{4} \sqrt {-b} b d \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) - 315 \, a^{4} \sqrt {-b} b d \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, {\left (35 \, a^{5} c x^{5} + a^{3} b^{2} c x^{3} - {\left (8 \, a b^{4} c - 315 \, a^{5} d\right )} x - {\left (35 \, a^{4} c x^{4} + 6 \, a^{2} b^{2} c x^{2} - 16 \, b^{4} c + 315 \, a^{4} d\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{315 \, a^{4} b^{2}}\right ] \]

[In]

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

[Out]

[1/315*(630*a^4*b^(3/2)*d*arctan(sqrt(a*x + sqrt(a^2*x^2 + b^2))/sqrt(b)) + 315*a^4*b^(3/2)*d*log((b^2 + sqrt(
a*x + sqrt(a^2*x^2 + b^2))*((a*x - b)*sqrt(b) - sqrt(a^2*x^2 + b^2)*sqrt(b)) + sqrt(a^2*x^2 + b^2)*b)/x) - 2*(
35*a^5*c*x^5 + a^3*b^2*c*x^3 - (8*a*b^4*c - 315*a^5*d)*x - (35*a^4*c*x^4 + 6*a^2*b^2*c*x^2 - 16*b^4*c + 315*a^
4*d)*sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqrt(a^2*x^2 + b^2)))/(a^4*b^2), 1/315*(630*a^4*sqrt(-b)*b*d*arctan(sqrt(
a*x + sqrt(a^2*x^2 + b^2))*sqrt(-b)/b) - 315*a^4*sqrt(-b)*b*d*log(-(b^2 + sqrt(a*x + sqrt(a^2*x^2 + b^2))*((a*
x + b)*sqrt(-b) - sqrt(a^2*x^2 + b^2)*sqrt(-b)) - sqrt(a^2*x^2 + b^2)*b)/x) - 2*(35*a^5*c*x^5 + a^3*b^2*c*x^3
- (8*a*b^4*c - 315*a^5*d)*x - (35*a^4*c*x^4 + 6*a^2*b^2*c*x^2 - 16*b^4*c + 315*a^4*d)*sqrt(a^2*x^2 + b^2))*sqr
t(a*x + sqrt(a^2*x^2 + b^2)))/(a^4*b^2)]

Sympy [F]

\[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {c x^{4} + d}{x \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

[In]

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

[Out]

Integral((c*x**4 + d)/(x*sqrt(a*x + sqrt(a**2*x**2 + b**2))), x)

Maxima [F]

\[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {c x^{4} + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x} \,d x } \]

[In]

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

[Out]

integrate((c*x^4 + d)/(sqrt(a*x + sqrt(a^2*x^2 + b^2))*x), x)

Giac [F]

\[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {c x^{4} + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x} \,d x } \]

[In]

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

[Out]

integrate((c*x^4 + d)/(sqrt(a*x + sqrt(a^2*x^2 + b^2))*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {c\,x^4+d}{x\,\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]

[In]

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

[Out]

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