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]

-2/3*e*(g*x+f)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(3/2)+4/3*e*g*(g*x+f)^(1/2)/(a*e^2+c*d^2)/(-d*g+e*f)/(e*x+d)^(1/2)+
e*(c*d*f+a*e*g-(-d*g+e*f)*(-a)^(1/2)*c^(1/2))*(g*x+f)^(1/2)/(a*e^2+c*d^2)/(-d*g+e*f)/(-a)^(1/2)/(e*(-a)^(1/2)+
d*c^(1/2))/(e*x+d)^(1/2)-e*(c*d*f+a*e*g+(-d*g+e*f)*(-a)^(1/2)*c^(1/2))*(g*x+f)^(1/2)/(a*e^2+c*d^2)/(-d*g+e*f)/
(-a)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))/(e*x+d)^(1/2)+arctanh((e*x+d)^(1/2)*(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+
f)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2))*c^(1/2)*(c*d*f+a*e*g+(-d*g+e*f)*(-a)^(1/2)*c^(1/2))/(a*e^2+c*d^2)/(-
a)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))^(3/2)/(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)+arctanh((e*x+d)^(1/2)*(g*(-a)^(1/2)+f
*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(1/2))*c^(1/2)*(c*d*f*(-a)^(1/2)+a*e*g*(-a)^(1/2)+a*(-d
*g+e*f)*c^(1/2))/a/(a*e^2+c*d^2)/(e*(-a)^(1/2)+d*c^(1/2))^(3/2)/(g*(-a)^(1/2)+f*c^(1/2))^(1/2)

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Rubi [A]  time = 3.16, antiderivative size = 613, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 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 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 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 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 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 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 {\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*}

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Mathematica [A]  time = 2.86, size = 353, normalized size = 0.58 \[ -\frac {2 e \sqrt {f+g x} \left (a e^3 (f+g x)+c d \left (-6 d^2 g+7 d e f-5 d e g x+6 e^2 f x\right )\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (e f-d g)}-\frac {\sqrt {c} \sqrt {\sqrt {-a} g-\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{\left (\sqrt {-a} e-\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d+a e\right )}-\frac {\sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d-a e\right )} \]

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]*ArcTanh[(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]*ArcTanh[(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))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [B]  time = 0.09, size = 14861, normalized size = 24.24 \[ \text {output too large to display} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g x + f}}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {f+g\,x}}{\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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