3.25.56 \(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [2456]

Optimal. Leaf size=716 \[ -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(1/2)-10/63*(-14*c*e*x-15*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)*(e*x+d)^(1/2)/e^3-2
/63*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(-36*a*e+37*b*d)-12*c^2*d*e*(-11*a*e+20*b*d)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*
e*(-7*a*e+8*b*d))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^5+2/63*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+64
*b*d)-b^2*c*e^3*(-15*a*e+7*b*d)+3*c^2*e^2*(28*a^2*e^2-76*a*b*d*e+45*b^2*d^2))*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)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^2/e^6/(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)-2/63*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(128*c^2*d^2-b^2*e^2-4
*c*e*(-33*a*e+32*b*d))*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)*(-4*a*c+b^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/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.78, antiderivative size = 716, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {746, 828, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\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 )}{63 c^2 e^6 \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 {2 \sqrt {2} \sqrt {b^2-4 a c} \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 ) \left (-4 c e (32 b d-33 a e)-b^2 e^2+128 c^2 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 )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{63 c e^5}-\frac {10 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*d*e*(20*b*d - 11*a*e) - 3*c*e*
(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*
d - 15*b*e - 14*c*e*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3) - (2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (2*
Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e)
+ 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*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)])/(63*c^2*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[
a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - b^2*e^2
- 4*c*e*(32*b*d - 33*a*e))*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)])/(63*c^2*e^6*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 746

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

Rule 828

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

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 {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {10 \int \frac {\left (\frac {1}{2} c \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\right )-\frac {1}{2} c \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e^3}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} c \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\right )\right )+\frac {1}{2} c \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{63 c^2 e^5}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (2 \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{63 c e^6}+\frac {\left (4 \left (-\frac {1}{2} c d \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right )+\frac {1}{4} c e \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{63 c^2 e^6}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\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 )}{63 c^2 e^6 \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 (8 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {1}{2} c d \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right )+\frac {1}{4} c e \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\right )\right )\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 )}{63 c^3 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\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 )}{63 c^2 e^6 \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 {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-128 b c d e-b^2 e^2+132 a c 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 )}{63 c^2 e^6 \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 = 32.93, size = 7946, normalized size = 11.10 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(9186\) vs. \(2(648)=1296\).
time = 0.89, size = 9187, normalized size = 12.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(2072\)
elliptic	\(\text {Expression too large to display}\)	\(2528\)
default	\(\text {Expression too large to display}\)	\(9187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),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((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(x*e + d)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.14, size = 1017, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left ({\left (256 \, c^{5} d^{6} - {\left (2 \, b^{5} - 33 \, a b^{3} c + 228 \, a^{2} b c^{2}\right )} x e^{6} - {\left ({\left (13 \, b^{4} c - 258 \, a b^{2} c^{2} - 456 \, a^{2} c^{3}\right )} d x + {\left (2 \, b^{5} - 33 \, a b^{3} c + 228 \, a^{2} b c^{2}\right )} d\right )} e^{5} - {\left ({\left (77 \, b^{3} c^{2} + 972 \, a b c^{3}\right )} d^{2} x + {\left (13 \, b^{4} c - 258 \, a b^{2} c^{2} - 456 \, a^{2} c^{3}\right )} d^{2}\right )} e^{4} + {\left (2 \, {\left (239 \, b^{2} c^{3} + 324 \, a c^{4}\right )} d^{3} x - {\left (77 \, b^{3} c^{2} + 972 \, a b c^{3}\right )} d^{3}\right )} e^{3} - 2 \, {\left (320 \, b c^{4} d^{4} x - {\left (239 \, b^{2} c^{3} + 324 \, a c^{4}\right )} d^{4}\right )} e^{2} + 128 \, {\left (2 \, c^{5} d^{5} x - 5 \, b c^{4} d^{5}\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 ) + 6 \, {\left (128 \, c^{5} d^{5} e - {\left (b^{4} c - 15 \, a b^{2} c^{2} - 84 \, a^{2} c^{3}\right )} x e^{6} - {\left ({\left (7 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d x + {\left (b^{4} c - 15 \, a b^{2} c^{2} - 84 \, a^{2} c^{3}\right )} d\right )} e^{5} + {\left (3 \, {\left (45 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{2} x - {\left (7 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d^{2}\right )} e^{4} - {\left (256 \, b c^{4} d^{3} x - 3 \, {\left (45 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{3}\right )} e^{3} + 128 \, {\left (c^{5} d^{4} x - 2 \, b c^{4} d^{4}\right )} e^{2}\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 (128 \, c^{5} d^{4} e^{2} - {\left (7 \, c^{5} x^{4} + 19 \, b c^{4} x^{3} - 63 \, a^{2} c^{3} + {\left (15 \, b^{2} c^{3} + 28 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} + 57 \, a b c^{3}\right )} x\right )} e^{6} + {\left (10 \, c^{5} d x^{3} + 31 \, b c^{4} d x^{2} + {\left (33 \, b^{2} c^{3} + 58 \, a c^{4}\right )} d x - {\left (b^{3} c^{2} + 183 \, a b c^{3}\right )} d\right )} e^{5} - {\left (16 \, c^{5} d^{2} x^{2} + 64 \, b c^{4} d^{2} x - {\left (111 \, b^{2} c^{3} + 212 \, a c^{4}\right )} d^{2}\right )} e^{4} + 16 \, {\left (2 \, c^{5} d^{3} x - 15 \, b c^{4} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{189 \, {\left (c^{3} x e^{8} + c^{3} d e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/189*((256*c^5*d^6 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*x*e^6 - ((13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*
d*x + (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d)*e^5 - ((77*b^3*c^2 + 972*a*b*c^3)*d^2*x + (13*b^4*c - 258*a*b^2*
c^2 - 456*a^2*c^3)*d^2)*e^4 + (2*(239*b^2*c^3 + 324*a*c^4)*d^3*x - (77*b^3*c^2 + 972*a*b*c^3)*d^3)*e^3 - 2*(32
0*b*c^4*d^4*x - (239*b^2*c^3 + 324*a*c^4)*d^4)*e^2 + 128*(2*c^5*d^5*x - 5*b*c^4*d^5)*e)*sqrt(c)*e^(1/2)*weiers
trassPInverse(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) + 6*(128*c^5*d^5
*e - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*x*e^6 - ((7*b^3*c^2 + 228*a*b*c^3)*d*x + (b^4*c - 15*a*b^2*c^2 - 84*a
^2*c^3)*d)*e^5 + (3*(45*b^2*c^3 + 76*a*c^4)*d^2*x - (7*b^3*c^2 + 228*a*b*c^3)*d^2)*e^4 - (256*b*c^4*d^3*x - 3*
(45*b^2*c^3 + 76*a*c^4)*d^3)*e^3 + 128*(c^5*d^4*x - 2*b*c^4*d^4)*e^2)*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*(128*c^5*d^4*e^2 - (7*c^5*x^4 + 19*b*c^4*x^3 - 63*a^2*c^3 + (15*b^2*c^3 + 28*a*
c^4)*x^2 + (b^3*c^2 + 57*a*b*c^3)*x)*e^6 + (10*c^5*d*x^3 + 31*b*c^4*d*x^2 + (33*b^2*c^3 + 58*a*c^4)*d*x - (b^3
*c^2 + 183*a*b*c^3)*d)*e^5 - (16*c^5*d^2*x^2 + 64*b*c^4*d^2*x - (111*b^2*c^3 + 212*a*c^4)*d^2)*e^4 + 16*(2*c^5
*d^3*x - 15*b*c^4*d^3)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c^3*x*e^8 + c^3*d*e^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(3/2), x)

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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((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(x*e + d)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2), x)

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