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

Optimal. Leaf size=236 \[ \frac {2 d}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {b^2 c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{12 a^4}+\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}{56 a^4}+\frac {b^8 c}{72 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \]

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Rubi [A]  time = 0.73, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6742, 2119, 453, 329, 298, 203, 206, 448} \begin {gather*} \frac {2 d}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[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 203

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

Rule 206

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

Rule 298

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 329

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 448

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 453

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 2119

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 6742

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

Rubi steps

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

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Mathematica [C]  time = 8.04, size = 915, normalized size = 3.88 \begin {gather*} \frac {2 \left (-1155 d \left (2 a x-\sqrt {b^2+a^2 x^2}\right ) \left (a x+\sqrt {b^2+a^2 x^2}\right ) a^4-\frac {1155 d \left (b^2+a^2 x^2\right ) \left (a x+\sqrt {b^2+a^2 x^2}\right )^2 \left (-2 b^2-a x \left (a x+\sqrt {b^2+a^2 x^2}\right )+2 \left (b^2+2 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}{b^2}\right )\right ) a^4}{\left (b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^2}+\frac {c \sqrt {b^2+a^2 x^2} \left (128 b^{10}+144 a x \left (9 a x+4 \sqrt {b^2+a^2 x^2}\right ) b^8+195 a^3 x^3 \left (9 a x+8 \sqrt {b^2+a^2 x^2}\right ) b^6-15 a^5 x^5 \left (97 a x+\sqrt {b^2+a^2 x^2}\right ) b^4-180 a^7 x^7 \left (37 a x+23 \sqrt {b^2+a^2 x^2}\right ) b^2-5040 a^9 x^9 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^3 \left (b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}+\frac {c \sqrt {b^2+a^2 x^2} \left (b^2+2 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^4 \left (-304 b^{10}-342 a x \left (9 a x+4 \sqrt {b^2+a^2 x^2}\right ) b^8-15 a^3 x^3 \left (249 a x+247 \sqrt {b^2+a^2 x^2}\right ) b^6+15 a^5 x^5 \left (317 a x+89 \sqrt {b^2+a^2 x^2}\right ) b^4+180 a^7 x^7 \left (59 a x+45 \sqrt {b^2+a^2 x^2}\right ) b^2+5040 a^9 x^9 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2048 a^{13} x^{13}+2048 a^{12} \sqrt {b^2+a^2 x^2} x^{12}+7168 a^{11} b^2 x^{11}+6144 a^{10} b^2 \sqrt {b^2+a^2 x^2} x^{10}+9728 a^9 b^4 x^9+6912 a^8 b^4 \sqrt {b^2+a^2 x^2} x^8+6400 a^7 b^6 x^7+3584 a^6 b^6 \sqrt {b^2+a^2 x^2} x^6+2072 a^5 b^8 x^5+840 a^4 b^8 \sqrt {b^2+a^2 x^2} x^4+292 a^3 b^{10} x^3+72 a^2 b^{10} \sqrt {b^2+a^2 x^2} x^2+12 a b^{12} x+b^{12} \sqrt {b^2+a^2 x^2}}\right )}{3465 a^4 b^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*a^4*d*(2*a*x - Sqrt[b^2 + a^2*x^2])*(a*x + Sqrt[b^2 + a^2*x^2]) + (c*Sqrt[b^2 + a^2*x^2]*(128*b^10 -
5040*a^9*x^9*(a*x + Sqrt[b^2 + a^2*x^2]) - 15*a^5*b^4*x^5*(97*a*x + Sqrt[b^2 + a^2*x^2]) + 144*a*b^8*x*(9*a*x
+ 4*Sqrt[b^2 + a^2*x^2]) + 195*a^3*b^6*x^3*(9*a*x + 8*Sqrt[b^2 + a^2*x^2]) - 180*a^7*b^2*x^7*(37*a*x + 23*Sqr
t[b^2 + a^2*x^2])))/((a*x + Sqrt[b^2 + a^2*x^2])^3*(b^2 + a*x*(a*x + Sqrt[b^2 + a^2*x^2]))) + (c*Sqrt[b^2 + a^
2*x^2]*(b^2 + 2*a*x*(a*x + Sqrt[b^2 + a^2*x^2]))^4*(-304*b^10 + 5040*a^9*x^9*(a*x + Sqrt[b^2 + a^2*x^2]) - 342
*a*b^8*x*(9*a*x + 4*Sqrt[b^2 + a^2*x^2]) + 180*a^7*b^2*x^7*(59*a*x + 45*Sqrt[b^2 + a^2*x^2]) + 15*a^5*b^4*x^5*
(317*a*x + 89*Sqrt[b^2 + a^2*x^2]) - 15*a^3*b^6*x^3*(249*a*x + 247*Sqrt[b^2 + a^2*x^2])))/(12*a*b^12*x + 292*a
^3*b^10*x^3 + 2072*a^5*b^8*x^5 + 6400*a^7*b^6*x^7 + 9728*a^9*b^4*x^9 + 7168*a^11*b^2*x^11 + 2048*a^13*x^13 + b
^12*Sqrt[b^2 + a^2*x^2] + 72*a^2*b^10*x^2*Sqrt[b^2 + a^2*x^2] + 840*a^4*b^8*x^4*Sqrt[b^2 + a^2*x^2] + 3584*a^6
*b^6*x^6*Sqrt[b^2 + a^2*x^2] + 6912*a^8*b^4*x^8*Sqrt[b^2 + a^2*x^2] + 6144*a^10*b^2*x^10*Sqrt[b^2 + a^2*x^2] +
2048*a^12*x^12*Sqrt[b^2 + a^2*x^2]) - (1155*a^4*d*(b^2 + a^2*x^2)*(a*x + Sqrt[b^2 + a^2*x^2])^2*(-2*b^2 - a*x
*(a*x + Sqrt[b^2 + a^2*x^2]) + 2*(b^2 + 2*a*x*(a*x + Sqrt[b^2 + a^2*x^2]))*Hypergeometric2F1[3/4, 1, 7/4, (a*x
+ Sqrt[b^2 + a^2*x^2])^2/b^2]))/(b^2 + a*x*(a*x + Sqrt[b^2 + a^2*x^2]))^2))/(3465*a^4*b^2*Sqrt[a*x + Sqrt[b^2
+ a^2*x^2]])

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IntegrateAlgebraic [A]  time = 0.47, size = 236, normalized size = 1.00 \begin {gather*} \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 \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]

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fricas [A]  time = 0.56, size = 482, normalized size = 2.04 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{4}+d}{x \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c\,x^4+d}{x\,\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{x \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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