3.17.49 \(\int \frac {(b+2 c x) (d+e x)^{7/2}}{(a+b x+c x^2)^{5/2}} \, dx\) [1649]

Optimal. Leaf size=573 \[ -\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {14 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )}+\frac {14 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {14 \sqrt {2} e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/3*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2)-14/3*e*(e*x+d)^(3/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x
+a)^(1/2)+14/3*e^2*(-b*e+2*c*d)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)+14/3*e*(c^2*d^2+b^2*e^2-c*e*(
3*a*e+b*d))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(
1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^2/
(-4*a*c+b^2)^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-14/3*e*(-b*e+2*c*d)*
(a*e^2-b*d*e+c*d^2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*
c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)
/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^2/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {782, 752, 846, 857, 732, 435, 430} \begin {gather*} -\frac {14 \sqrt {2} e \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {14 \sqrt {2} e \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {14 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c \left (b^2-4 a c\right )}-\frac {14 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(7/2))/(3*(a + b*x + c*x^2)^(3/2)) - (14*e*(d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(
b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (14*e^2*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c*(b^2 - 4
*a*c)) + (14*Sqrt[2]*e*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2
- 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 -
4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (14*Sqrt[2]*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e
*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(
b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 -
4*a*c])*e)])/(3*c^2*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 752

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

Rule 782

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

Rule 846

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

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} (7 e) \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {(14 e) \int \frac {\sqrt {d+e x} \left (-\frac {3}{2} e (b d-2 a e)-\frac {3}{2} e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {14 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )}-\frac {(28 e) \int \frac {-\frac {3}{4} e \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )-\frac {3}{2} e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{9 c \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {14 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )}-\frac {\left (7 e (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )}+\frac {\left (14 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {14 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )}+\frac {\left (14 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (14 \sqrt {2} e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (d+e x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {14 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {14 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )}+\frac {14 \sqrt {2} e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {14 \sqrt {2} e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 31.29, size = 1259, normalized size = 2.20 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {2 \left (7 b^3 e^3 x^2+7 b e \left (a^2 e^2+c^2 d x^2 (3 d-2 e x)+a c \left (d^2-6 d e x-e^2 x^2\right )\right )+b^2 \left (14 a e^3 x+c \left (d^3+10 d^2 e x-11 d e^2 x^2+8 e^3 x^3\right )\right )-2 c \left (-7 c^2 d^2 e x^3+7 a^2 e^2 (2 d+e x)+a c \left (2 d^3-d^2 e x+20 d e^2 x^2+9 e^3 x^3\right )\right )\right )}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {7 \left (4 e^2 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (-3 a^2 c e^2+\left (c^2 d^2-b c d e+b^2 e^2\right ) x (b+c x)+a \left (b^2 e^2-b c e (d+3 e x)+c^2 \left (d^2-3 e^2 x^2\right )\right )\right )-i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-i \sqrt {2} \left (b^3 e^3-b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (-4 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (-c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{c^2 \left (b^2-4 a c\right ) \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (d+e x)}\right )}{3 \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[d + e*x]*((2*(7*b^3*e^3*x^2 + 7*b*e*(a^2*e^2 + c^2*d*x^2*(3*d - 2*e*x) + a*c*(d^2 - 6*d*e*x - e^2*x^2))
+ b^2*(14*a*e^3*x + c*(d^3 + 10*d^2*e*x - 11*d*e^2*x^2 + 8*e^3*x^3)) - 2*c*(-7*c^2*d^2*e*x^3 + 7*a^2*e^2*(2*d
+ e*x) + a*c*(2*d^3 - d^2*e*x + 20*d*e^2*x^2 + 9*e^3*x^3))))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (7*(4*e^2*
Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(-3*a^2*c*e^2 + (c^2*d^2 - b*c*d*e +
b^2*e^2)*x*(b + c*x) + a*(b^2*e^2 - b*c*e*(d + 3*e*x) + c^2*(d^2 - 3*e^2*x^2))) - I*Sqrt[2]*(2*c*d - b*e + Sq
rt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 -
4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]
)*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d +
e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e +
a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/
(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - I*Sqrt[2]*(b^3*e^3 - b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b
*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2
- 4*a*c)*e^2])))*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c
)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4
*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2
])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^
2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(
c^2*(b^2 - 4*a*c)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(d + e*x))))/(3*Sq
rt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12974\) vs. \(2(503)=1006\).
time = 0.96, size = 12975, normalized size = 22.64

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\left (\frac {2 \left (a c \,e^{2}-b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e x}{3 c^{4}}-\frac {2 \left (a b \,e^{3}-3 a d \,e^{2} c +c^{2} d^{3}\right )}{3 c^{4}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (\frac {2 \left (9 a c \,e^{2}-4 b^{2} e^{2}+7 b c d e -7 c^{2} d^{2}\right ) e x}{3 c^{2} \left (4 a c -b^{2}\right )}-\frac {\left (11 a b c \,e^{2}-40 a \,c^{2} d e -b^{3} e^{2}+3 b^{2} d c e +7 d^{2} b \,c^{2}\right ) e}{3 c^{3} \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (c e x +c d \right )}}+\frac {2 \left (-\frac {e^{3} \left (3 b e -8 c d \right )}{c^{2}}+\frac {2 \left (20 c \,e^{3} a b -58 d \,e^{2} c^{2} a -5 b^{3} e^{3}+18 b^{2} d \,e^{2} c -14 b \,c^{2} d^{2} e +14 c^{3} d^{3}\right ) e}{3 c^{2} \left (4 a c -b^{2}\right )}-\frac {e^{2} \left (11 a b c \,e^{2}-40 a \,c^{2} d e -b^{3} e^{2}+3 b^{2} d c e +7 d^{2} b \,c^{2}\right )}{3 c^{2} \left (4 a c -b^{2}\right )}+\frac {4 d \left (9 a c \,e^{2}-4 b^{2} e^{2}+7 b c d e -7 c^{2} d^{2}\right ) e}{3 c \left (4 a c -b^{2}\right )}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}+\frac {2 \left (\frac {2 e^{4}}{c}+\frac {2 \left (9 a c \,e^{2}-4 b^{2} e^{2}+7 b c d e -7 c^{2} d^{2}\right ) e^{2}}{3 c \left (4 a c -b^{2}\right )}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(1265\)
default	\(\text {Expression too large to display}\)	\(12975\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*(x*e + d)^(7/2)/(c*x^2 + b*x + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.25, size = 1266, normalized size = 2.21 \begin {gather*} -\frac {2 \, {\left (7 \, {\left (2 \, c^{5} d^{3} x^{4} + 4 \, b c^{4} d^{3} x^{3} + 4 \, a b c^{3} d^{3} x + 2 \, a^{2} c^{3} d^{3} + 2 \, {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{3} x^{2} + {\left (2 \, a^{2} b^{3} - 9 \, a^{3} b c + {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} x^{4} + 2 \, {\left (2 \, b^{4} c - 9 \, a b^{2} c^{2}\right )} x^{3} + {\left (2 \, b^{5} - 5 \, a b^{3} c - 18 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left (2 \, a b^{4} - 9 \, a^{2} b^{2} c\right )} x\right )} e^{3} - 3 \, {\left ({\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d x^{4} + 2 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{3} + {\left (b^{4} c - 4 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d x + {\left (a^{2} b^{2} c - 6 \, a^{3} c^{2}\right )} d\right )} e^{2} - 3 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} d^{2} x + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 42 \, {\left ({\left (a^{2} b^{2} c - 3 \, a^{3} c^{2} + {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} x^{4} + 2 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c - a b^{2} c^{2} - 6 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c - 3 \, a^{2} b c^{2}\right )} x\right )} e^{3} - {\left (b c^{4} d x^{4} + 2 \, b^{2} c^{3} d x^{3} + 2 \, a b^{2} c^{2} d x + a^{2} b c^{2} d + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d x^{2}\right )} e^{2} + {\left (c^{5} d^{2} x^{4} + 2 \, b c^{4} d^{2} x^{3} + 2 \, a b c^{3} d^{2} x + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} + {\left (7 \, a^{2} b c^{2} + 2 \, {\left (4 \, b^{2} c^{3} - 9 \, a c^{4}\right )} x^{3} + 7 \, {\left (b^{3} c^{2} - a b c^{3}\right )} x^{2} + 14 \, {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} x\right )} e^{3} - {\left (14 \, b c^{4} d x^{3} + 42 \, a b c^{3} d x + 28 \, a^{2} c^{3} d + {\left (11 \, b^{2} c^{3} + 40 \, a c^{4}\right )} d x^{2}\right )} e^{2} + {\left (14 \, c^{5} d^{2} x^{3} + 21 \, b c^{4} d^{2} x^{2} + 7 \, a b c^{3} d^{2} + 2 \, {\left (5 \, b^{2} c^{3} + a c^{4}\right )} d^{2} x\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{9 \, {\left (a^{2} b^{2} c^{3} - 4 \, a^{3} c^{4} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + {\left (b^{4} c^{3} - 2 \, a b^{2} c^{4} - 8 \, a^{2} c^{5}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} - 4 \, a^{2} b c^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

-2/9*(7*(2*c^5*d^3*x^4 + 4*b*c^4*d^3*x^3 + 4*a*b*c^3*d^3*x + 2*a^2*c^3*d^3 + 2*(b^2*c^3 + 2*a*c^4)*d^3*x^2 + (
2*a^2*b^3 - 9*a^3*b*c + (2*b^3*c^2 - 9*a*b*c^3)*x^4 + 2*(2*b^4*c - 9*a*b^2*c^2)*x^3 + (2*b^5 - 5*a*b^3*c - 18*
a^2*b*c^2)*x^2 + 2*(2*a*b^4 - 9*a^2*b^2*c)*x)*e^3 - 3*((b^2*c^3 - 6*a*c^4)*d*x^4 + 2*(b^3*c^2 - 6*a*b*c^3)*d*x
^3 + (b^4*c - 4*a*b^2*c^2 - 12*a^2*c^3)*d*x^2 + 2*(a*b^3*c - 6*a^2*b*c^2)*d*x + (a^2*b^2*c - 6*a^3*c^2)*d)*e^2
- 3*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + 2*a*b^2*c^2*d^2*x + a^2*b*c^2*d^2 + (b^3*c^2 + 2*a*b*c^3)*d^2*x^2)*e
)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3
- 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^
(-1)/c) + 42*((a^2*b^2*c - 3*a^3*c^2 + (b^2*c^3 - 3*a*c^4)*x^4 + 2*(b^3*c^2 - 3*a*b*c^3)*x^3 + (b^4*c - a*b^2*
c^2 - 6*a^2*c^3)*x^2 + 2*(a*b^3*c - 3*a^2*b*c^2)*x)*e^3 - (b*c^4*d*x^4 + 2*b^2*c^3*d*x^3 + 2*a*b^2*c^2*d*x + a
^2*b*c^2*d + (b^3*c^2 + 2*a*b*c^3)*d*x^2)*e^2 + (c^5*d^2*x^4 + 2*b*c^4*d^2*x^3 + 2*a*b*c^3*d^2*x + a^2*c^3*d^2
+ (b^2*c^3 + 2*a*c^4)*d^2*x^2)*e)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)
*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3,
weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e
- 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) + 3*((b^
2*c^3 - 4*a*c^4)*d^3 + (7*a^2*b*c^2 + 2*(4*b^2*c^3 - 9*a*c^4)*x^3 + 7*(b^3*c^2 - a*b*c^3)*x^2 + 14*(a*b^2*c^2
- a^2*c^3)*x)*e^3 - (14*b*c^4*d*x^3 + 42*a*b*c^3*d*x + 28*a^2*c^3*d + (11*b^2*c^3 + 40*a*c^4)*d*x^2)*e^2 + (14
*c^5*d^2*x^3 + 21*b*c^4*d^2*x^2 + 7*a*b*c^3*d^2 + 2*(5*b^2*c^3 + a*c^4)*d^2*x)*e)*sqrt(c*x^2 + b*x + a)*sqrt(x
*e + d))/(a^2*b^2*c^3 - 4*a^3*c^4 + (b^2*c^5 - 4*a*c^6)*x^4 + 2*(b^3*c^4 - 4*a*b*c^5)*x^3 + (b^4*c^3 - 2*a*b^2
*c^4 - 8*a^2*c^5)*x^2 + 2*(a*b^3*c^3 - 4*a^2*b*c^4)*x)

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Sympy [F(-1)] Timed out
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((2*c*x+b)*(e*x+d)**(7/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*x + b)*(x*e + d)^(7/2)/(c*x^2 + b*x + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^(5/2),x)

[Out]

int(((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^(5/2), x)

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