3.2285 \(\int (d+e x)^m (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=221 \[ \frac{(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]

[Out]

-((g*(d + e*x)^m*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(c*e^2*(7 + m))) + ((2*c*d - b*e)^2*(b*e*g*(7 +
2*m) - 2*c*(d*g*m + e*f*(7 + m)))*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b*e))^(-1/2 - m)*(c*d - b*e - c*e*x)^3*S
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]*Hypergeometric2F1[7/2, -5/2 - m, 9/2, (c*d - b*e - c*e*x)/(2*c*d - b*
e)])/(7*c^4*e^2*(7 + m))

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Rubi [A]  time = 0.352932, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {794, 679, 677, 70, 69} \[ \frac{(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

-((g*(d + e*x)^m*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(c*e^2*(7 + m))) + ((2*c*d - b*e)^2*(b*e*g*(7 +
2*m) - 2*c*(d*g*m + e*f*(7 + m)))*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b*e))^(-1/2 - m)*(c*d - b*e - c*e*x)^3*S
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]*Hypergeometric2F1[7/2, -5/2 - m, 9/2, (c*d - b*e - c*e*x)/(2*c*d - b*
e)])/(7*c^4*e^2*(7 + m))

Rule 794

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

Rule 679

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^IntPart[m]*(d + e*
x)^FracPart[m])/(1 + (e*x)/d)^FracPart[m], Int[(1 + (e*x)/d)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ
[d, 0])

Rule 677

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^m*(a + b*x + c*x^2
)^FracPart[p])/((1 + (e*x)/d)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)
/e)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))

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 (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac{(b e g (7+2 m)-2 c (d g m+e f (7+m))) \int (d+e x)^m \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{2 c e (7+m)}\\ &=-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac{\left ((b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-m}\right ) \int \left (1+\frac{e x}{d}\right )^m \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{2 c e (7+m)}\\ &=-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac{\left ((b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (1+\frac{e x}{d}\right )^{-\frac{1}{2}-m} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}\right ) \int \left (1+\frac{e x}{d}\right )^{\frac{5}{2}+m} \left (c d^2-b d e-c d e x\right )^{5/2} \, dx}{2 c e (7+m) \sqrt{c d^2-b d e-c d e x}}\\ &=-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac{\left (\left (-c d e-\frac{e \left (c d^2-b d e\right )}{d}\right )^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (-\frac{c d e \left (1+\frac{e x}{d}\right )}{-c d e-\frac{e \left (c d^2-b d e\right )}{d}}\right )^{-\frac{1}{2}-m} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}\right ) \int \left (c d^2-b d e-c d e x\right )^{5/2} \left (\frac{c d}{2 c d-b e}+\frac{c e x}{2 c d-b e}\right )^{\frac{5}{2}+m} \, dx}{2 c^3 d^2 e^3 (7+m) \sqrt{c d^2-b d e-c d e x}}\\ &=-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}+\frac{(2 c d-b e)^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-\frac{1}{2}-m} (c d-b e-c e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \, _2F_1\left (\frac{7}{2},-\frac{5}{2}-m;\frac{9}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.754468, size = 161, normalized size = 0.73 \[ \frac{(d+e x)^{m-3} ((d+e x) (c (d-e x)-b e))^{7/2} \left (-(b e-2 c d)^2 \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (2 c (d g m+e f (m+7))-b e g (2 m+7)) \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )-7 c^3 g (d+e x)^3\right )}{7 c^4 e^2 (m+7)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

((d + e*x)^(-3 + m)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(7/2)*(-7*c^3*g*(d + e*x)^3 - (-2*c*d + b*e)^2*(-(b*e*g
*(7 + 2*m)) + 2*c*(d*g*m + e*f*(7 + m)))*((c*(d + e*x))/(2*c*d - b*e))^(-1/2 - m)*Hypergeometric2F1[7/2, -5/2
- m, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]))/(7*c^4*e^2*(7 + m))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

int((e*x+d)^m*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (g x + f\right )}{\left (e x + d\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)*(e*x + d)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{2} e^{4} g x^{5} +{\left (c^{2} e^{4} f + 2 \, b c e^{4} g\right )} x^{4} +{\left (2 \, b c e^{4} f -{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} - b^{2} e^{4}\right )} g\right )} x^{3} -{\left ({\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} - b^{2} e^{4}\right )} f + 2 \,{\left (b c d^{2} e^{2} - b^{2} d e^{3}\right )} g\right )} x^{2} +{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} f -{\left (2 \,{\left (b c d^{2} e^{2} - b^{2} d e^{3}\right )} f -{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e x + d\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

integral((c^2*e^4*g*x^5 + (c^2*e^4*f + 2*b*c*e^4*g)*x^4 + (2*b*c*e^4*f - (2*c^2*d^2*e^2 - 2*b*c*d*e^3 - b^2*e^
4)*g)*x^3 - ((2*c^2*d^2*e^2 - 2*b*c*d*e^3 - b^2*e^4)*f + 2*(b*c*d^2*e^2 - b^2*d*e^3)*g)*x^2 + (c^2*d^4 - 2*b*c
*d^3*e + b^2*d^2*e^2)*f - (2*(b*c*d^2*e^2 - b^2*d*e^3)*f - (c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*g)*x)*sqrt(-c
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^m, 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)**m*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (g x + f\right )}{\left (e x + d\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)*(e*x + d)^m, x)