3.17.36 \(\int \frac {(b+2 c x) (a+b x+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [1636]

Optimal. Leaf size=701 \[ \frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/15*(16*c^2*d^3+3*a*b*e^3-c*d*e*(-4*a*e+13*b*d)+e*(22*c^2*d^2+3*b^2*e^2-2*c*e*(-5*a*e+11*b*d))*x)*(c*x^2+b*x
+a)^(3/2)/e^2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)+2/15*c*(128*c^2*d^3-4*c*d*e*(-29*a*e+44*b*d)+3*b*e^2*(-16*a*e+
17*b*d)+e*(32*c^2*d^2+3*b^2*e^2-4*c*e*(-5*a*e+8*b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^4/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(
1/2)-1/15*(-b*e+2*c*d)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(-29*a*e+32*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
)*(-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)/e^5/(a*e^2-b*d*e+c*d^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)+4/15*(128*c^2*d^2+27*b^2*e^2-4*c*e*(-5*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)/e^5/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {824, 826, 857, 732, 435, 430} \begin {gather*} \frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-5 a e)+27 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 )}{15 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\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}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 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 )}{15 e^5 \sqrt {a+b x+c x^2} \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 )}}}+\frac {2 c \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{15 e^4 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(2*c*(128*c^2*d^3 - 4*c*d*e*(44*b*d - 29*a*e) + 3*b*e^2*(17*b*d - 16*a*e) + e*(32*c^2*d^2 + 3*b^2*e^2 - 4*c*e*
(8*b*d - 5*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*e^4*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (2*(16*c^2*d^3 + 3
*a*b*e^3 - c*d*e*(13*b*d - 4*a*e) + e*(22*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(11*b*d - 5*a*e))*x)*(a + b*x + c*x^2)^(
3/2))/(15*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2
+ 3*b^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[A
rcSin[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)])/(15*e^5*(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[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*a*e))*S
qrt[(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)])/(15*e^5*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 824

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

Rule 826

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m +
2*p + 2))), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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 {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} c \left (16 b c d^2-13 b^2 d e-12 a c d e+16 a b e^2\right )-\frac {1}{2} c \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (51 b^3 d e^2-8 a c e \left (8 c d^2+5 a e^2\right )+4 b c d \left (32 c d^2+45 a e^2\right )-2 b^2 \left (88 c d^2 e+27 a e^3\right )\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 e^5 \left (c d^2-b d e+a e^2\right )}+\frac {\left (2 c \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^5}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{15 e^5 \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 {a+b x+c x^2}}+\frac {\left (4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\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 )}{15 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{15 e^5 \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 {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 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 )}{15 e^5 \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 = 16.49, size = 5450, normalized size = 7.77 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(7/2),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. \(19269\) vs. \(2(637)=1274\).
time = 1.04, size = 19270, normalized size = 27.49

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1427\)
risch	\(\text {Expression too large to display}\)	\(4808\)
default	\(\text {Expression too large to display}\)	\(19270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),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)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(x*e + d)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.96, size = 1380, normalized size = 1.97 \begin {gather*} \frac {2 \, {\left ({\left (256 \, c^{4} d^{7} - {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} x^{3} e^{7} - {\left (2 \, {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d x^{3} + 3 \, {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d x^{2}\right )} e^{6} + {\left (2 \, {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{2} x^{3} - 6 \, {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{2} x^{2} - 3 \, {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{2} x\right )} e^{5} - {\left (512 \, b c^{3} d^{3} x^{3} - 6 \, {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{3} x^{2} + 6 \, {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{3} x + {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} + 2 \, {\left (128 \, c^{4} d^{4} x^{3} - 768 \, b c^{3} d^{4} x^{2} + 3 \, {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{4} x - {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{4}\right )} e^{3} + 2 \, {\left (384 \, c^{4} d^{5} x^{2} - 768 \, b c^{3} d^{5} x + {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{5}\right )} e^{2} + 256 \, {\left (3 \, c^{4} d^{6} x - 2 \, b c^{3} d^{6}\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 ) + 3 \, {\left (256 \, c^{4} d^{6} e - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} x^{3} e^{7} + {\left (2 \, {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d x^{3} - 3 \, {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d x^{2}\right )} e^{6} - 3 \, {\left (128 \, b c^{3} d^{2} x^{3} - 2 \, {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{2} x^{2} + {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d^{2} x\right )} e^{5} + {\left (256 \, c^{4} d^{3} x^{3} - 1152 \, b c^{3} d^{3} x^{2} + 6 \, {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{3} x - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d^{3}\right )} e^{4} + 2 \, {\left (384 \, c^{4} d^{4} x^{2} - 576 \, b c^{3} d^{4} x + {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{4}\right )} e^{3} + 384 \, {\left (2 \, c^{4} d^{5} x - b c^{3} d^{5}\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^{4} d^{5} e^{2} + {\left (10 \, a c^{3} x^{3} - 3 \, a^{2} b c - {\left (3 \, b^{3} c + 61 \, a b c^{2}\right )} x^{2} - 2 \, {\left (3 \, a b^{2} c + 5 \, a^{2} c^{2}\right )} x\right )} e^{7} - {\left (10 \, b c^{3} d x^{3} + 78 \, a b c^{2} d x + 4 \, a^{2} c^{2} d - {\left (79 \, b^{2} c^{2} + 152 \, a c^{3}\right )} d x^{2}\right )} e^{6} + {\left (10 \, c^{4} d^{2} x^{3} - 249 \, b c^{3} d^{2} x^{2} - 35 \, a b c^{2} d^{2} + 2 \, {\left (59 \, b^{2} c^{2} + 115 \, a c^{3}\right )} d^{2} x\right )} e^{5} + {\left (176 \, c^{4} d^{3} x^{2} - 400 \, b c^{3} d^{3} x + {\left (51 \, b^{2} c^{2} + 100 \, a c^{3}\right )} d^{3}\right )} e^{4} + 16 \, {\left (18 \, c^{4} d^{4} x - 11 \, b c^{3} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{2} d^{5} e^{6} + a c x^{3} e^{11} - {\left (b c d x^{3} - 3 \, a c d x^{2}\right )} e^{10} + {\left (c^{2} d^{2} x^{3} - 3 \, b c d^{2} x^{2} + 3 \, a c d^{2} x\right )} e^{9} + {\left (3 \, c^{2} d^{3} x^{2} - 3 \, b c d^{3} x + a c d^{3}\right )} e^{8} + {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*((256*c^4*d^7 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*x^3*e^7 - (2*(11*b^3*c + 212*a*b*c^2)*d*x^3 + 3*(3*b^4
- 46*a*b^2*c - 120*a^2*c^2)*d*x^2)*e^6 + (2*(139*b^2*c^2 + 212*a*c^3)*d^2*x^3 - 6*(11*b^3*c + 212*a*b*c^2)*d^
2*x^2 - 3*(3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*d^2*x)*e^5 - (512*b*c^3*d^3*x^3 - 6*(139*b^2*c^2 + 212*a*c^3)*d^3
*x^2 + 6*(11*b^3*c + 212*a*b*c^2)*d^3*x + (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*d^3)*e^4 + 2*(128*c^4*d^4*x^3 - 7
68*b*c^3*d^4*x^2 + 3*(139*b^2*c^2 + 212*a*c^3)*d^4*x - (11*b^3*c + 212*a*b*c^2)*d^4)*e^3 + 2*(384*c^4*d^5*x^2
- 768*b*c^3*d^5*x + (139*b^2*c^2 + 212*a*c^3)*d^5)*e^2 + 256*(3*c^4*d^6*x - 2*b*c^3*d^6)*e)*sqrt(c)*e^(1/2)*we
ierstrassPInverse(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*(256*c^4
*d^6*e - (3*b^3*c + 116*a*b*c^2)*x^3*e^7 + (2*(67*b^2*c^2 + 116*a*c^3)*d*x^3 - 3*(3*b^3*c + 116*a*b*c^2)*d*x^2
)*e^6 - 3*(128*b*c^3*d^2*x^3 - 2*(67*b^2*c^2 + 116*a*c^3)*d^2*x^2 + (3*b^3*c + 116*a*b*c^2)*d^2*x)*e^5 + (256*
c^4*d^3*x^3 - 1152*b*c^3*d^3*x^2 + 6*(67*b^2*c^2 + 116*a*c^3)*d^3*x - (3*b^3*c + 116*a*b*c^2)*d^3)*e^4 + 2*(38
4*c^4*d^4*x^2 - 576*b*c^3*d^4*x + (67*b^2*c^2 + 116*a*c^3)*d^4)*e^3 + 384*(2*c^4*d^5*x - b*c^3*d^5)*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^4*d^5*e^2 + (10*a*c^3*x^3 - 3*a^2*b*c
- (3*b^3*c + 61*a*b*c^2)*x^2 - 2*(3*a*b^2*c + 5*a^2*c^2)*x)*e^7 - (10*b*c^3*d*x^3 + 78*a*b*c^2*d*x + 4*a^2*c^
2*d - (79*b^2*c^2 + 152*a*c^3)*d*x^2)*e^6 + (10*c^4*d^2*x^3 - 249*b*c^3*d^2*x^2 - 35*a*b*c^2*d^2 + 2*(59*b^2*c
^2 + 115*a*c^3)*d^2*x)*e^5 + (176*c^4*d^3*x^2 - 400*b*c^3*d^3*x + (51*b^2*c^2 + 100*a*c^3)*d^3)*e^4 + 16*(18*c
^4*d^4*x - 11*b*c^3*d^4)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c^2*d^5*e^6 + a*c*x^3*e^11 - (b*c*d*x^3 -
3*a*c*d*x^2)*e^10 + (c^2*d^2*x^3 - 3*b*c*d^2*x^2 + 3*a*c*d^2*x)*e^9 + (3*c^2*d^3*x^2 - 3*b*c*d^3*x + a*c*d^3)*
e^8 + (3*c^2*d^4*x - b*c*d^4)*e^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Integral((b + 2*c*x)*(a + b*x + c*x**2)**(3/2)/(d + e*x)**(7/2), 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((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(x*e + d)^(7/2), 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 (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(7/2),x)

[Out]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(7/2), x)

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