\(\int (d+e x)^m (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [2285]

Optimal result

Integrand size = 44, antiderivative size = 221 \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}+\frac {(2 c d-b e)^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} (c d-b e-c e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (7+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {808, 693, 691, 72, 71} \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m-\frac {5}{2},\frac {9}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]

[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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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]))

Rule 72

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 691

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/e)
*x)^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 693

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 808

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])

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.91 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.73 \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x)^{-3+m} ((d+e x) (-b e+c (d-e x)))^{7/2} \left (-7 c^3 g (d+e x)^3-(-2 c d+b e)^2 (-b e g (7+2 m)+2 c (d g m+e f (7+m))) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {-c d+b e+c e x}{-2 c d+b e}\right )\right )}{7 c^4 e^2 (7+m)} \]

[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))

Maple [F]

\[\int \left (e x +d \right )^{m} \left (g x +f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}d x\]

[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)

Fricas [F]

\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]

[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)

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[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]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]

[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)

Giac [F]

\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2} \,d x \]

[In]

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

[Out]

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