3.17.32 \(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [1632]

Optimal. Leaf size=691 \[ \frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/15*(8*c^2*d^3-c*d*e*(-4*a*e+5*b*d)-b*e^2*(-3*a*e+2*b*d)+e*(14*c^2*d^2+b^2*e^2-2*c*e*(-5*a*e+7*b*d))*x)*(c*x
^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)+4/15*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))
*(c*x^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)-2/15*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*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)*(-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^3/(a*e^2-b*d*e+c*d^2)^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)+2/
15*(16*c^2*d^2-b^2*e^2-4*c*e*(-5*a*e+4*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^3/(a*e^2-b*d*e+c*d^2)/(
e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 691, 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, 848, 857, 732, 435, 430} \begin {gather*} \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}} \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 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^3 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\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}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 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^3 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{15 e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 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)*Sqrt[a + b*x + c*x^2])/(d + e*x)^(7/2),x]

[Out]

(4*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*Sqrt[a + b*x + c*x^2])/(15*e^2*(c*d^2 - b*d*e + a
*e^2)^2*Sqrt[d + e*x]) - (2*(8*c^2*d^3 - c*d*e*(5*b*d - 4*a*e) - b*e^2*(2*b*d - 3*a*e) + e*(14*c^2*d^2 + b^2*e
^2 - 2*c*e*(7*b*d - 5*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (2*Sq
rt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*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)])/(15*e^3*(c*d^2 - b*d*e + a*e^2)^2*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]*(16*c^2*d
^2 - b^2*e^2 - 4*c*e*(4*b*d - 5*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)])/(15*e^3*(c*d^2 - b*d*e + a*e^2)*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 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (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) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} \left (5 b^2 c d e+12 a c^2 d e+2 b^3 e^2-8 b c \left (c d^2+2 a e^2\right )\right )-\frac {1}{2} c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^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 (b^3 d e^2-4 a c e \left (c d^2+5 a e^2\right )+4 b c d \left (2 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )\right )-\frac {1}{2} c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {\left (c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 e^3 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 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^3 \left (c d^2-b d e+a e^2\right )^2 \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 (2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2-b^2 e^2-4 c e (4 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^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 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^3 \left (c d^2-b d e+a e^2\right )^2 \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} \left (16 c^2 d^2-b^2 e^2-4 c e (4 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^3 \left (c d^2-b d e+a e^2\right ) \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 = 16.38, size = 5427, normalized size = 7.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^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. \(19264\) vs. \(2(627)=1254\).
time = 1.14, size = 19265, normalized size = 27.88

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (10 a c \,e^{2}+b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{15 e^{4} \left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 \left (c e \,x^{2}+b e x +a e \right ) \left (8 c \,e^{3} a b -16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right )}{15 e^{3} \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 \left (\frac {2 c^{2}}{e^{3}}-\frac {c \left (10 a c \,e^{2}+b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{15 e^{3} \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {2 \left (b e -c d \right ) \left (8 c \,e^{3} a b -16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right )}{15 e^{3} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {2 b \left (8 c \,e^{3} a b -16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right )}{15 e^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\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 {4 c \left (8 c \,e^{3} a b -16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\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 )}{15 e^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \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}}\)	\(1311\)
default	\(\text {Expression too large to display}\)	\(19265\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(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)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(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 = 0.55, size = 1487, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{4} d^{7} + {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} x^{3} e^{7} + {\left ({\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d x^{3} + 3 \, {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d x^{2}\right )} e^{6} + {\left ({\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{2} x^{3} + 3 \, {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d^{2} x\right )} e^{5} - {\left (32 \, b c^{3} d^{3} x^{3} - 3 \, {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{3} x^{2} - 3 \, {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{3} x - {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} + {\left (16 \, c^{4} d^{4} x^{3} - 96 \, b c^{3} d^{4} x^{2} + 3 \, {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{4} x + {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{4}\right )} e^{3} + {\left (48 \, c^{4} d^{5} x^{2} - 96 \, b c^{3} d^{5} x + {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{5}\right )} e^{2} + 16 \, {\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 ) + 6 \, {\left (8 \, c^{4} d^{6} e + {\left (b^{3} c - 8 \, a b c^{2}\right )} x^{3} e^{7} + {\left (2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d x^{3} + 3 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} d x^{2}\right )} e^{6} - 3 \, {\left (4 \, b c^{3} d^{2} x^{3} - 2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{2} - {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{2} x\right )} e^{5} + {\left (8 \, c^{4} d^{3} x^{3} - 36 \, b c^{3} d^{3} x^{2} + 6 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{3} x + {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{3}\right )} e^{4} + 2 \, {\left (12 \, c^{4} d^{4} x^{2} - 18 \, b c^{3} d^{4} x + {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{4}\right )} e^{3} + 12 \, {\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 (8 \, c^{4} d^{5} e^{2} - {\left (3 \, a^{2} b c - 2 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} x^{2} + {\left (a b^{2} c + 10 \, a^{2} c^{2}\right )} x\right )} e^{7} + {\left (4 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d x^{2} + {\left (5 \, b^{3} c - 8 \, a b c^{2}\right )} d x + {\left (5 \, a b^{2} c - 4 \, a^{2} c^{2}\right )} d\right )} e^{6} - {\left (24 \, b c^{3} d^{2} x^{2} + 10 \, a b c^{2} d^{2} + {\left (7 \, b^{2} c^{2} - 40 \, a c^{3}\right )} d^{2} x\right )} e^{5} + {\left (16 \, c^{4} d^{3} x^{2} - 20 \, b c^{3} d^{3} x + {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} d^{3}\right )} e^{4} + {\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^{3} d^{7} e^{4} + a^{2} c x^{3} e^{11} - {\left (2 \, a b c d x^{3} - 3 \, a^{2} c d x^{2}\right )} e^{10} - {\left (6 \, a b c d^{2} x^{2} - 3 \, a^{2} c d^{2} x - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{3}\right )} e^{9} - {\left (2 \, b c^{2} d^{3} x^{3} + 6 \, a b c d^{3} x - a^{2} c d^{3} - 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{3} x^{2}\right )} e^{8} + {\left (c^{3} d^{4} x^{3} - 6 \, b c^{2} d^{4} x^{2} - 2 \, a b c d^{4} + 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x\right )} e^{7} + {\left (3 \, c^{3} d^{5} x^{2} - 6 \, b c^{2} d^{5} x + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{5}\right )} e^{6} + {\left (3 \, c^{3} d^{6} x - 2 \, b c^{2} d^{6}\right )} e^{5}\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)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*((16*c^4*d^7 + (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*x^3*e^7 + ((3*b^3*c - 44*a*b*c^2)*d*x^3 + 3*(2*b^4 - 19*
a*b^2*c + 60*a^2*c^2)*d*x^2)*e^6 + ((13*b^2*c^2 + 44*a*c^3)*d^2*x^3 + 3*(3*b^3*c - 44*a*b*c^2)*d^2*x^2 + 3*(2*
b^4 - 19*a*b^2*c + 60*a^2*c^2)*d^2*x)*e^5 - (32*b*c^3*d^3*x^3 - 3*(13*b^2*c^2 + 44*a*c^3)*d^3*x^2 - 3*(3*b^3*c
- 44*a*b*c^2)*d^3*x - (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*d^3)*e^4 + (16*c^4*d^4*x^3 - 96*b*c^3*d^4*x^2 + 3*(13
*b^2*c^2 + 44*a*c^3)*d^4*x + (3*b^3*c - 44*a*b*c^2)*d^4)*e^3 + (48*c^4*d^5*x^2 - 96*b*c^3*d^5*x + (13*b^2*c^2
+ 44*a*c^3)*d^5)*e^2 + 16*(3*c^4*d^6*x - 2*b*c^3*d^6)*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) + 6*(8*c^4*d^6*e + (b^3*c - 8*a*b*c^2)*x^3*e^7
+ (2*(b^2*c^2 + 8*a*c^3)*d*x^3 + 3*(b^3*c - 8*a*b*c^2)*d*x^2)*e^6 - 3*(4*b*c^3*d^2*x^3 - 2*(b^2*c^2 + 8*a*c^3)
*d^2*x^2 - (b^3*c - 8*a*b*c^2)*d^2*x)*e^5 + (8*c^4*d^3*x^3 - 36*b*c^3*d^3*x^2 + 6*(b^2*c^2 + 8*a*c^3)*d^3*x +
(b^3*c - 8*a*b*c^2)*d^3)*e^4 + 2*(12*c^4*d^4*x^2 - 18*b*c^3*d^4*x + (b^2*c^2 + 8*a*c^3)*d^4)*e^3 + 12*(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, weierstrass
PInverse(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*(8*c^4*d^5*e^2 -
(3*a^2*b*c - 2*(b^3*c - 8*a*b*c^2)*x^2 + (a*b^2*c + 10*a^2*c^2)*x)*e^7 + (4*(b^2*c^2 + 8*a*c^3)*d*x^2 + (5*b^
3*c - 8*a*b*c^2)*d*x + (5*a*b^2*c - 4*a^2*c^2)*d)*e^6 - (24*b*c^3*d^2*x^2 + 10*a*b*c^2*d^2 + (7*b^2*c^2 - 40*a
*c^3)*d^2*x)*e^5 + (16*c^4*d^3*x^2 - 20*b*c^3*d^3*x + (b^2*c^2 + 20*a*c^3)*d^3)*e^4 + (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^3*d^7*e^4 + a^2*c*x^3*e^11 - (2*a*b*c*d*x^3 - 3*a^2*c*d*x^2
)*e^10 - (6*a*b*c*d^2*x^2 - 3*a^2*c*d^2*x - (b^2*c + 2*a*c^2)*d^2*x^3)*e^9 - (2*b*c^2*d^3*x^3 + 6*a*b*c*d^3*x
- a^2*c*d^3 - 3*(b^2*c + 2*a*c^2)*d^3*x^2)*e^8 + (c^3*d^4*x^3 - 6*b*c^2*d^4*x^2 - 2*a*b*c*d^4 + 3*(b^2*c + 2*a
*c^2)*d^4*x)*e^7 + (3*c^3*d^5*x^2 - 6*b*c^2*d^5*x + (b^2*c + 2*a*c^2)*d^5)*e^6 + (3*c^3*d^6*x - 2*b*c^2*d^6)*e
^5)

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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 ) \sqrt {a + b x + c x^{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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**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)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(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 )\,\sqrt {c\,x^2+b\,x+a}}{{\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)^(1/2))/(d + e*x)^(7/2),x)

[Out]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(7/2), x)

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