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

Optimal. Leaf size=161 \[ \frac{2 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.196042, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104, Rules used = {866, 891, 63, 217, 206} \[ \frac{2 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e*g*n)/(c*(p + 1)), I
nt[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] &
& LtQ[p, -1] && GtQ[n, 0]

Rule 891

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*
(f + g*x)^n*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2
- 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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])

Rubi steps

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

Mathematica [A]  time = 0.39543, size = 176, normalized size = 1.09 \[ \frac{2 \sqrt{d+e x} \left (\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x} \sqrt{c d f-a e g} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )-(c d)^{3/2} (f+g x)\right )}{(c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.389, size = 210, normalized size = 1.3 \begin{align*}{\frac{1}{d \left ( cdx+ae \right ) c}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( \ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg} \right ){\frac{1}{\sqrt{cdg}}}} \right ) xcdg+\ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg} \right ){\frac{1}{\sqrt{cdg}}}} \right ) aeg-2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg} \right ){\frac{1}{\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }}}{\frac{1}{\sqrt{cdg}}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 6.66954, size = 1227, normalized size = 7.62 \begin{align*} \left [\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{\frac{g}{c d}} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + 4 \,{\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} \sqrt{\frac{g}{c d}} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{2 \,{\left (c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}, -\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-\frac{g}{c d}} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} c d \sqrt{-\frac{g}{c d}}}{2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(g/(c*d))*log(-(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^
2*e*f*g + a^2*d*e^2*g^2 + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + 4*(2*c^2*d^2*g*x + c^2*d^2*f + a
*c*d*e*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(g/(c*d)) + (c^2*d^2*e*f
^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)) - 4*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(c^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x), -((c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-g/(c*d))*arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e
*x + d)*sqrt(g*x + f)*c*d*sqrt(-g/(c*d))/(2*c*d*e*g*x^2 + c*d^2*f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*
x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(c^2*d^2*e*x^2 + a*c*d^2*e +
(c^2*d^3 + a*c*d*e^2)*x)]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x