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

Optimal. Leaf size=549 \[ -\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}+\frac {g \sqrt {d+e x} \left (2 \sqrt {-a} e g-\sqrt {c} (d g+e f)\right )}{\sqrt {-a} \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2}+\frac {g \sqrt {d+e x} \left (2 \sqrt {-a} e g+\sqrt {c} (d g+e f)\right )}{\sqrt {-a} \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\sqrt {-a} g+\sqrt {c} f\right ) (e f-d g)^2}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}} \]

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Rubi [A]  time = 1.32, antiderivative size = 543, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {912, 104, 152, 12, 93, 208} \begin {gather*} -\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e}{\sqrt {-a} \sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}+\frac {g \sqrt {d+e x} \left (2 a e g-\sqrt {-a} \sqrt {c} (d g+e f)\right )}{a \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\sqrt {-a} g+\sqrt {c} f\right ) (e f-d g)^2}+\frac {g \sqrt {d+e x} \left (\sqrt {-a} \sqrt {c} (d g+e f)+2 a e g\right )}{a \sqrt {f+g x} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2}+\frac {c \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 (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {c \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 )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(e/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(e*f - d*g)*Sqrt[d + e*x]*Sqrt[f + g*x])) + e/(Sqrt[-a]*(Sqrt[c]*d + Sq
rt[-a]*e)*(e*f - d*g)*Sqrt[d + e*x]*Sqrt[f + g*x]) + (g*(2*a*e*g - Sqrt[-a]*Sqrt[c]*(e*f + d*g))*Sqrt[d + e*x]
)/(a*(Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)^2*Sqrt[f + g*x]) + (g*(2*a*e*g + Sqrt[-a]*S
qrt[c]*(e*f + d*g))*Sqrt[d + e*x])/(a*(Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)^2*Sqrt[f +
g*x]) + (c*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)*(Sqrt[c]*f - Sqrt[-a]*g)^(3/2)) - (c*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)
*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 104

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 912

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}}\\ &=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}-\frac {\int \frac {\frac {1}{2} \left (2 \sqrt {-a} e g+\sqrt {c} (e f-d g)\right )+\sqrt {c} e g x}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {\int \frac {\frac {1}{2} \left (2 \sqrt {-a} e g-\sqrt {c} (e f-d g)\right )-\sqrt {c} e g x}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g)}\\ &=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \int -\frac {c (e f-d g)^2}{4 \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2}+\frac {2 \int -\frac {c (e f-d g)^2}{4 \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2}\\ &=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}-\frac {c \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 (\sqrt {c} f+\sqrt {-a} g\right )}-\frac {c \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)\right )}\\ &=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}-\frac {c \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 (\sqrt {c} f+\sqrt {-a} g\right )}-\frac {c \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} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)}\\ &=-\frac {e}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x} \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g-\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {g \left (2 \sqrt {-a} e g+\sqrt {c} (e f+d g)\right ) \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g)^2 \sqrt {f+g x}}+\frac {c \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 {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (\sqrt {-a} c d f+(-a)^{3/2} e g+a \sqrt {c} (e f+d g)\right )}-\frac {c \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 (\sqrt {c} f+\sqrt {-a} g\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 2.07, size = 521, normalized size = 0.95 \begin {gather*} \frac {\frac {e}{\sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {-a} e-\sqrt {c} d\right )}+\frac {e}{\sqrt {d+e x} \sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {g \sqrt {d+e x} \left (2 \sqrt {-a} e g+\sqrt {c} (d g+e f)\right )}{\sqrt {f+g x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\sqrt {-a} g+\sqrt {c} f\right ) (e f-d g)}+\frac {\frac {g \sqrt {d+e x} \left (2 \sqrt {-a} e g-\sqrt {c} (d g+e f)\right )}{\sqrt {f+g x} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)}+\frac {c (e f-d g) \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 )}{\sqrt {\sqrt {-a} e-\sqrt {c} d} \left (\sqrt {-a} g-\sqrt {c} f\right )^{3/2}}}{\sqrt {c} d-\sqrt {-a} e}+\frac {c (d g-e 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} g+\sqrt {c} f\right )^{3/2}}}{\sqrt {-a} (e f-d g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e/((-(Sqrt[c]*d) + Sqrt[-a]*e)*Sqrt[d + e*x]*Sqrt[f + g*x]) + e/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*Sqrt[
f + g*x]) + (g*(2*Sqrt[-a]*e*g + Sqrt[c]*(e*f + d*g))*Sqrt[d + e*x])/((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[c]*f + Sq
rt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + ((g*(2*Sqrt[-a]*e*g - Sqrt[c]*(e*f + d*g))*Sqrt[d + e*x])/((Sqrt[c]*f -
Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + (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[-(Sqrt[c]*d) + Sqrt[-a]*e]*(-(Sqrt[c]*f) + Sqrt[-a]
*g)^(3/2)))/(Sqrt[c]*d - Sqrt[-a]*e) + (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[c]*d + Sqrt[-a]*e)^(3/2)*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))
)/(Sqrt[-a]*(e*f - d*g))

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IntegrateAlgebraic [C]  time = 1.69, size = 492, normalized size = 0.90 \begin {gather*} -\frac {i c \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \tan ^{-1}\left (\frac {\sqrt {f+g x} \sqrt {a e^2+c d^2}}{\sqrt {d+e x} \sqrt {-i \sqrt {a} \sqrt {c} d g+i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^{3/2} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+\frac {i c \left (\sqrt {c} d+i \sqrt {a} e\right )^2 \tan ^{-1}\left (\frac {\sqrt {f+g x} \sqrt {a e^2+c d^2}}{\sqrt {d+e x} \sqrt {i \sqrt {a} \sqrt {c} d g-i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^{3/2} \left (\sqrt {c} f-i \sqrt {a} g\right ) \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}-\frac {2 \sqrt {d+e x} \left (\frac {a e^3 g^2 (f+g x)}{d+e x}+a e^2 g^3+c d^2 g^3+\frac {c e^3 f^2 (f+g x)}{d+e x}\right )}{\sqrt {f+g x} \left (a e^2+c d^2\right ) \left (a g^2+c f^2\right ) (d g-e f)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(c*d^2*g^3 + a*e^2*g^3 + (c*e^3*f^2*(f + g*x))/(d + e*x) + (a*e^3*g^2*(f + g*x))/(d + e*x)))
/((c*d^2 + a*e^2)*(-(e*f) + d*g)^2*(c*f^2 + a*g^2)*Sqrt[f + g*x]) - (I*c*(Sqrt[c]*d - I*Sqrt[a]*e)^2*ArcTan[(S
qrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[-(c*d*f) + I*Sqrt[a]*Sqrt[c]*e*f - I*Sqrt[a]*Sqrt[c]*d*g - a*e*g]*Sqrt
[d + e*x])])/(Sqrt[a]*(c*d^2 + a*e^2)^(3/2)*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c
]*f + I*Sqrt[a]*g))]) + (I*c*(Sqrt[c]*d + I*Sqrt[a]*e)^2*ArcTan[(Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[-(c*
d*f) - I*Sqrt[a]*Sqrt[c]*e*f + I*Sqrt[a]*Sqrt[c]*d*g - a*e*g]*Sqrt[d + e*x])])/(Sqrt[a]*(c*d^2 + a*e^2)^(3/2)*
(Sqrt[c]*f - I*Sqrt[a]*g)*Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]*f - I*Sqrt[a]*g))])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(g*x+f)^(3/2)/(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 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.23, size = 30656, normalized size = 55.84 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(g*x+f)^(3/2)/(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 \begin {gather*} \int \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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