3.609 \(\int \frac{\sqrt{f+g x}}{(d+e x)^{5/2} (a+c x^2)} \, dx\)

Optimal. Leaf size=613 \[ \frac{e \sqrt{f+g x} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right )}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) (e f-d g)}-\frac{e \sqrt{f+g x} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right )}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right ) (e f-d g)}+\frac{4 e g \sqrt{f+g x}}{3 \sqrt{d+e x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac{2 e \sqrt{f+g x}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac{\sqrt{c} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\sqrt{c} \left (a \sqrt{c} (e f-d g)+\sqrt{-a} c d f+\sqrt{-a} a e g\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]

[Out]

(-2*e*Sqrt[f + g*x])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (4*e*g*Sqrt[f + g*x])/(3*(c*d^2 + a*e^2)*(e*f - d*g
)*Sqrt[d + e*x]) + (e*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f - d*g))*Sqrt[f + g*x])/(Sqrt[-a]*(Sqrt[c]*d + Sqr
t[-a]*e)*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[d + e*x]) - (e*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*Sqrt[f
+ g*x])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[d + e*x]) + (Sqrt[c]*(c*d*f + a*e
*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt
[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)^(3/2)*(c*d^2 + a*e^2)*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]
) + (Sqrt[c]*(Sqrt[-a]*c*d*f + Sqrt[-a]*a*e*g + a*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*S
qrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*(Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*(c*d^2 + a*e^2)
*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

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Rubi [A]  time = 3.1602, antiderivative size = 613, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {908, 45, 37, 6725, 96, 93, 208} \[ \frac{e \sqrt{f+g x} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right )}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) (e f-d g)}-\frac{e \sqrt{f+g x} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right )}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right ) (e f-d g)}+\frac{4 e g \sqrt{f+g x}}{3 \sqrt{d+e x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac{2 e \sqrt{f+g x}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac{\sqrt{c} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\sqrt{c} \left (a \sqrt{c} (e f-d g)+\sqrt{-a} c d f+\sqrt{-a} a e g\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[f + g*x]/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e*Sqrt[f + g*x])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (4*e*g*Sqrt[f + g*x])/(3*(c*d^2 + a*e^2)*(e*f - d*g
)*Sqrt[d + e*x]) + (e*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f - d*g))*Sqrt[f + g*x])/(Sqrt[-a]*(Sqrt[c]*d + Sqr
t[-a]*e)*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[d + e*x]) - (e*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*Sqrt[f
+ g*x])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[d + e*x]) + (Sqrt[c]*(c*d*f + a*e
*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt
[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)^(3/2)*(c*d^2 + a*e^2)*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]
) + (Sqrt[c]*(Sqrt[-a]*c*d*f + Sqrt[-a]*a*e*g + a*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*S
qrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*(Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*(c*d^2 + a*e^2)
*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

Rule 908

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

Rule 45

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{\sqrt{f+g x}}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx &=\frac{\int \frac{c d f+a e g-c (e f-d g) x}{(d+e x)^{3/2} \sqrt{f+g x} \left (a+c x^2\right )} \, dx}{c d^2+a e^2}+\frac{(e (e f-d g)) \int \frac{1}{(d+e x)^{5/2} \sqrt{f+g x}} \, dx}{c d^2+a e^2}\\ &=-\frac{2 e \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac{\int \left (\frac{a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}}+\frac{-a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}}\right ) \, dx}{c d^2+a e^2}-\frac{(2 e g) \int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac{2 e \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac{4 e g \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c d^2+a e^2\right )}-\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c d^2+a e^2\right )}\\ &=-\frac{2 e \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac{4 e g \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{e \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{\left (\sqrt{c} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{c} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right )}\\ &=-\frac{2 e \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac{4 e g \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{e \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{\left (\sqrt{c} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c} d+\sqrt{-a} e-\left (\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{c} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{c} d+\sqrt{-a} e-\left (-\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right )}\\ &=-\frac{2 e \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac{4 e g \sqrt{f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}-\frac{e \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{d+e x}}+\frac{\sqrt{c} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \left (c d^2+a e^2\right ) \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{c} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f+\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right )^{3/2} \left (c d^2+a e^2\right ) \sqrt{\sqrt{c} f+\sqrt{-a} g}}\\ \end{align*}

Mathematica [A]  time = 4.22092, size = 355, normalized size = 0.58 \[ -\frac{2 e \sqrt{f+g x} \left (a e^3 (f+g x)+c d \left (-6 d^2 g+7 d e f-5 d e g x+6 e^2 f x\right )\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (e f-d g)}+\frac{\sqrt{c} \sqrt{-\sqrt{-a} g-\sqrt{c} f} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (\sqrt{-a} \sqrt{c} d-a e\right )}+\frac{\sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\left (\sqrt{-a} e-\sqrt{c} d\right )^{3/2} \left (\sqrt{-a} \sqrt{c} d+a e\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[f + g*x]/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e*Sqrt[f + g*x]*(a*e^3*(f + g*x) + c*d*(7*d*e*f - 6*d^2*g + 6*e^2*f*x - 5*d*e*g*x)))/(3*(c*d^2 + a*e^2)^2*
(e*f - d*g)*(d + e*x)^(3/2)) + (Sqrt[c]*Sqrt[-(Sqrt[c]*f) - Sqrt[-a]*g]*ArcTan[(Sqrt[-(Sqrt[c]*f) - Sqrt[-a]*g
]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*(Sqrt[-a]*Sqrt
[c]*d - a*e)) + (Sqrt[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*ArcTan[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqr
t[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-(Sqrt[c]*d) + Sqrt[-a]*e)^(3/2)*(Sqrt[-a]*Sqrt[c]*d + a*e))

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Maple [B]  time = 0.458, size = 14861, normalized size = 24.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^(1/2)/(e*x+d)^(5/2)/(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{\sqrt{g x + f}}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(g*x + f)/((c*x^2 + a)*(e*x + d)^(5/2)), x)

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Fricas [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((g*x+f)^(1/2)/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((g*x+f)**(1/2)/(e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="giac")

[Out]

Exception raised: TypeError