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

Optimal. Leaf size=213 \[ \frac{\sqrt{d+e x} (f+g x)^{n+1} (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \, _2F_1\left (1,n+\frac{3}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{c d g (n+1) (2 n+3) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}+\frac{2 e (f+g x)^{n+1} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt{d+e x}} \]

[Out]

(2*e*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*g*(3 + 2*n)*Sqrt[d + e*x]) + ((2*a*e^
2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e + c*d*x)*Sqrt[d + e*x]*(f + g*x)^(1 + n)*Hypergeometric2F1[1, 3/
2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(c*d*g*(c*d*f - a*e*g)*(1 + n)*(3 + 2*n)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.278404, antiderivative size = 222, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {880, 891, 70, 69} \[ \frac{2 e (f+g x)^{n+1} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^n (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (2 n+3) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*g*(3 + 2*n)*Sqrt[d + e*x]) - (2*(2*a*
e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e + c*d*x)*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[1/2, -n,
3/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))])/(c^2*d^2*g*(3 + 2*n)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 880

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(b*e*g*(
n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x +
c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && Eq
Q[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1] && IntegerQ[2*p]

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 70

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

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \frac{(d+e x)^{3/2} (f+g x)^n}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 e (f+g x)^{1+n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt{d+e x}}-\frac{\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \int \frac{\sqrt{d+e x} (f+g x)^n}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d g (3+2 n)}\\ &=\frac{2 e (f+g x)^{1+n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt{d+e x}}-\frac{\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \sqrt{a e+c d x} \sqrt{d+e x}\right ) \int \frac{(f+g x)^n}{\sqrt{a e+c d x}} \, dx}{c d g (3+2 n) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{2 e (f+g x)^{1+n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt{d+e x}}-\frac{\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \sqrt{a e+c d x} \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n}\right ) \int \frac{\left (\frac{c d f}{c d f-a e g}+\frac{c d g x}{c d f-a e g}\right )^n}{\sqrt{a e+c d x}} \, dx}{c d g (3+2 n) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{2 e (f+g x)^{1+n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt{d+e x}}-\frac{2 \left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) (a e+c d x) \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (3+2 n) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.135152, size = 145, normalized size = 0.68 \[ \frac{(f+g x)^n \sqrt{(d+e x) (a e+c d x)} \left (\left (c d (d g (2 n+3)-e f)-2 a e^2 g (n+1)\right ) \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )+c d e (f+g x)\right )}{c^2 d^2 g \left (n+\frac{3}{2}\right ) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^n*(c*d*e*(f + g*x) + ((-2*a*e^2*g*(1 + n) + c*d*(-(e*f) + d*g*(3 + 2*
n)))*Hypergeometric2F1[1/2, -n, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*(f + g*x))/(c*d*f - a*e*g))^
n))/(c^2*d^2*g*(3/2 + n)*Sqrt[d + e*x])

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Maple [F]  time = 1.633, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{n} \left ( ex+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}{\left (g x + f\right )}^{n}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*(g*x + f)^n/(c*d*x + a*e), 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)**n/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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