3.622 \(\int \frac{1}{(d+e x)^{3/2} (a+c x^2)} \, dx\)

Optimal. Leaf size=663 \[ -\frac{2 e}{\sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

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

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Rubi [A]  time = 0.968651, antiderivative size = 663, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {710, 827, 1169, 634, 618, 206, 628} \[ -\frac{2 e}{\sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)),x]

[Out]

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

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 827

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

Rule 1169

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

Rule 634

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.13075, size = 132, normalized size = 0.2 \[ \frac{\frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{-a} \sqrt{c} d+a e}}{\sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)),x]

[Out]

(-(Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) +
Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e))/Sqr
t[d + e*x]

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Maple [B]  time = 0.232, size = 5659, normalized size = 8.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*(e*x + d)^(3/2)), x)

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Fricas [B]  time = 2.48171, size = 5651, normalized size = 8.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

-1/2*((c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 +
3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^
2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 +
3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2
- 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*
d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d
^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*
c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15
*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2
*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d
*e^2 + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^
2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a
^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))*log(-(3*c^2*d^2*e - a
*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 -
a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8
*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 +
(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^
6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d
^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))) + (c*d^3 + a*d*e^2 + (c*d^
2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqr
t(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^
4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d
^2*e^4 + a^4*e^6))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 - (a*c^4*d^8
+ 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^
12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^
7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*
c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*
d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4
+ a^4*e^6))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 - (a*c^3*d^6 + 3*a^2*c^2*
d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c
^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a
*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) - (6*a*c
^2*d^3*e^2 - 2*a^2*c*d*e^4 - (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2
- 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15
*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^
2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^1
0*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d
^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))) + 4*sqrt(e*x + d)*e)/(c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3
)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a),x)

[Out]

Integral(1/((a + c*x**2)*(d + e*x)**(3/2)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a),x, algorithm="giac")

[Out]

Timed out