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

Optimal. Leaf size=196 \[ a d \text {RootSum}\left [\text {$\#$1}^8 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )\right )-b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 (-c)+2 \text {$\#$1} a^2 d+\text {$\#$1} b^2 c}\& \right ]-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \]

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Rubi [B]  time = 1.51, antiderivative size = 454, normalized size of antiderivative = 2.32, number of steps used = 21, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1166, 208, 205} \begin {gather*} \frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 205

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

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]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2117

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

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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.52, size = 190, normalized size = 0.97 \begin {gather*} a d \text {RootSum}\left [\text {$\#$1}^8 c-4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\&,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{\text {$\#$1}^5 c-2 \text {$\#$1} a^2 d-\text {$\#$1} b^2 c}\&\right ]-\frac {2 \left (b^2-a x \left (\sqrt {a^2 x^2+b^2}+a x\right )\right )}{3 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^2 - a*x*(a*x + Sqrt[b^2 + a^2*x^2])))/(3*a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + a*d*RootSum[b^4*c - 2*b^2
*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] + Log[Sqrt[a*x + Sqrt[b^2 +
a^2*x^2]] - #1]*#1^4)/(-(b^2*c*#1) - 2*a^2*d*#1 + c*#1^5) & ]

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IntegrateAlgebraic [A]  time = 0.38, size = 196, normalized size = 1.00 \begin {gather*} -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) + a*d*RootSum[b^4*c - 2*b
^2*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[
b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1 + 2*a^2*d*#1 - c*#1^5) & ]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> the translation of the FriCAS object sage2 to sage is not yet impleme
nted

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{d-c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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