3.67 \(\int \frac {(a+b x)^{3/2} \sqrt {c+d x} (A+B x+C x^2)}{\sqrt {e+f x}} \, dx\)

Optimal. Leaf size=1235 \[ \frac {2 C (c+d x)^{3/2} \sqrt {e+f x} (a+b x)^{5/2}}{9 b d f}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (c+d x)^{3/2} \sqrt {e+f x} (a+b x)^{3/2}}{63 b d^2 f^2}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {a+b x}}{315 b d^3 f^3}-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {a+b x}}{945 b^2 d^3 f^4}+\frac {2 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (2 C \left (92 d^3 e^3-33 c d^2 f e^2-18 c^2 d f^2 e-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3\right )-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2/63*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e))*(b*x+a)^(3/2)*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/d^2/f^2+2/9*C*(b*x+
a)^(5/2)*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/d/f-2/315*(7*b*d*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b*c*e)-(-3*a*d*f+4
*b*c*f+6*b*d*e)*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e)))*(d*x+c)^(3/2)*(b*x+a)^(1/2)*(f*x+e)^(1/2)/b/d^3/f^3-
2/945*(5*b*d*f*(7*a*d*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b*c*e)-(a*c*f+3*a*d*e+3*b*c*e)*(4*a*C*d*f+b*(-9*B*d*
f+6*C*c*f+8*C*d*e)))+2*(1/2*a*d*f-b*(c*f+2*d*e))*(7*b*d*f*(-9*A*b*d*f+C*a*c*f+3*C*a*d*e+5*C*b*c*e)-(-3*a*d*f+4
*b*c*f+6*b*d*e)*(4*a*C*d*f+b*(-9*B*d*f+6*C*c*f+8*C*d*e))))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/d^3/f
^4+2/315*(8*a^4*C*d^4*f^4+a^3*b*d^3*f^3*(-18*B*d*f-7*C*c*f+11*C*d*e)-3*a^2*b^2*d^2*f^2*(3*d*f*(-7*A*d*f-3*B*c*
f+4*B*d*e)-C*(-3*c^2*f^2-5*c*d*e*f+9*d^2*e^2))-a*b^3*d*f*(2*C*(-16*c^3*f^3-18*c^2*d*e*f^2-33*c*d^2*e^2*f+92*d^
3*e^3)+3*d*f*(7*A*d*f*(-7*c*f+13*d*e)-B*(-19*c^2*f^2-29*c*d*e*f+72*d^2*e^2)))+b^4*(C*(-16*c^4*f^4-16*c^3*d*e*f
^3-21*c^2*d^2*e^2*f^2-40*c*d^3*e^3*f+128*d^4*e^4)+3*d*f*(7*A*d*f*(-2*c^2*f^2-3*c*d*e*f+8*d^2*e^2)-B*(-8*c^3*f^
3-9*c^2*d*e*f^2-16*c*d^2*e^2*f+48*d^3*e^3))))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/
(-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^3/d^(7/2)/f^5/(d*x+c)^(1/2)/(b
*(f*x+e)/(-a*f+b*e))^(1/2)+2/315*(-a*f+b*e)*(-c*f+d*e)*(4*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(-3*B*d*f-C*c*f+3*C*d*
e)-3*a*b^2*d*f*(3*d*f*(-21*A*d*f+3*B*c*f+16*B*d*e)-5*C*(c^2*f^2+2*c*d*e*f+8*d^2*e^2))-b^3*(C*(8*c^3*f^3+15*c^2
*d*e*f^2+24*c*d^2*e^2*f+128*d^3*e^3)+3*d*f*(7*A*d*f*(c*f+8*d*e)-4*B*(c^2*f^2+2*c*d*e*f+12*d^2*e^2))))*Elliptic
F(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*
c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^3/d^(7/2)/f^5/(d*x+c)^(1/2)/(f*x+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 4.40, antiderivative size = 1235, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1615, 154, 158, 114, 113, 121, 120} \[ \frac {2 C (c+d x)^{3/2} \sqrt {e+f x} (a+b x)^{5/2}}{9 b d f}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (c+d x)^{3/2} \sqrt {e+f x} (a+b x)^{3/2}}{63 b d^2 f^2}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {a+b x}}{315 b d^3 f^3}-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {a+b x}}{945 b^2 d^3 f^4}+\frac {2 \sqrt {a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (2 C \left (92 d^3 e^3-33 c d^2 f e^2-18 c^2 d f^2 e-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]

[Out]

(-2*(5*b*d*f*(7*a*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (3*b*c*e + 3*a*d*e + a*c*f)*(4*a*C*d*f +
b*(8*C*d*e + 6*c*C*f - 9*B*d*f))) + 2*((a*d*f)/2 - b*(2*d*e + c*f))*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f
- 9*A*b*d*f) - (6*b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))))*Sqrt[a + b*x]*Sq
rt[c + d*x]*Sqrt[e + f*x])/(945*b^2*d^3*f^4) - (2*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (6*
b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f)))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e
+ f*x])/(315*b*d^3*f^3) - (2*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))*(a + b*x)^(3/2)*(c + d*x)^(3/2)*Sq
rt[e + f*x])/(63*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(9*b*d*f) + (2*Sqrt[-(b*c) +
a*d]*(8*a^4*C*d^4*f^4 + a^3*b*d^3*f^3*(11*C*d*e - 7*c*C*f - 18*B*d*f) - 3*a^2*b^2*d^2*f^2*(3*d*f*(4*B*d*e - 3
*B*c*f - 7*A*d*f) - C*(9*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) - a*b^3*d*f*(2*C*(92*d^3*e^3 - 33*c*d^2*e^2*f - 18*
c^2*d*e*f^2 - 16*c^3*f^3) + 3*d*f*(7*A*d*f*(13*d*e - 7*c*f) - B*(72*d^2*e^2 - 29*c*d*e*f - 19*c^2*f^2))) + b^4
*(C*(128*d^4*e^4 - 40*c*d^3*e^3*f - 21*c^2*d^2*e^2*f^2 - 16*c^3*d*e*f^3 - 16*c^4*f^4) + 3*d*f*(7*A*d*f*(8*d^2*
e^2 - 3*c*d*e*f - 2*c^2*f^2) - B*(48*d^3*e^3 - 16*c*d^2*e^2*f - 9*c^2*d*e*f^2 - 8*c^3*f^3))))*Sqrt[(b*(c + d*x
))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d
*(b*e - a*f))])/(315*b^3*d^(7/2)*f^5*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(b
*e - a*f)*(d*e - c*f)*(4*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(3*C*d*e - c*C*f - 3*B*d*f) - 3*a*b^2*d*f*(3*d*f*(16*
B*d*e + 3*B*c*f - 21*A*d*f) - 5*C*(8*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) - b^3*(C*(128*d^3*e^3 + 24*c*d^2*e^2*f +
15*c^2*d*e*f^2 + 8*c^3*f^3) + 3*d*f*(7*A*d*f*(8*d*e + c*f) - 4*B*(12*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*Sqrt[(b
*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c)
+ a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(315*b^3*d^(7/2)*f^5*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx &=\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {2 \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (-\frac {1}{2} b (5 b c C e+3 a C d e+a c C f-9 A b d f)-\frac {1}{2} b (4 a C d f+b (8 C d e+6 c C f-9 B d f)) x\right )}{\sqrt {e+f x}} \, dx}{9 b^2 d f}\\ &=-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {4 \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-\frac {1}{4} b (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-\frac {1}{4} b (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) x\right )}{\sqrt {e+f x}} \, dx}{63 b^2 d^2 f^2}\\ &=-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {8 \int \frac {\sqrt {c+d x} \left (-\frac {1}{8} b (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))))-\frac {1}{8} b \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) x\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx}{315 b^2 d^3 f^3}\\ &=-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{945 b^2 d^3 f^4}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {16 \int \frac {-\frac {1}{16} b \left (3 b c f (5 a d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))-(b c e+3 a d e+a c f) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))))-(b c e+a d e+a c f) \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right )\right )+\frac {3}{16} b \left (8 a^4 C d^4 f^4+a^3 b d^3 f^3 (11 C d e-7 c C f-18 B d f)-3 a^2 b^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )-a b^3 d f \left (2 C \left (92 d^3 e^3-33 c d^2 e^2 f-18 c^2 d e f^2-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d e f-19 c^2 f^2\right )\right )\right )+b^4 \left (C \left (128 d^4 e^4-40 c d^3 e^3 f-21 c^2 d^2 e^2 f^2-16 c^3 d e f^3-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 e^2 f-9 c^2 d e f^2-8 c^3 f^3\right )\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{945 b^3 d^3 f^4}\\ &=-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{945 b^2 d^3 f^4}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {\left ((b e-a f) (d e-c f) \left (4 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (3 C d e-c C f-3 B d f)-3 a b^2 d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d e f+c^2 f^2\right )\right )-b^3 \left (C \left (128 d^3 e^3+24 c d^2 e^2 f+15 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{315 b^2 d^3 f^5}+\frac {\left (8 a^4 C d^4 f^4+a^3 b d^3 f^3 (11 C d e-7 c C f-18 B d f)-3 a^2 b^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )-a b^3 d f \left (C \left (184 d^3 e^3-66 c d^2 e^2 f-36 c^2 d e f^2-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d e f-19 c^2 f^2\right )\right )\right )+b^4 \left (C \left (128 d^4 e^4-40 c d^3 e^3 f-21 c^2 d^2 e^2 f^2-16 c^3 d e f^3-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 e^2 f-9 c^2 d e f^2-8 c^3 f^3\right )\right )\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{315 b^2 d^3 f^5}\\ &=-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{945 b^2 d^3 f^4}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {\left ((b e-a f) (d e-c f) \left (4 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (3 C d e-c C f-3 B d f)-3 a b^2 d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d e f+c^2 f^2\right )\right )-b^3 \left (C \left (128 d^3 e^3+24 c d^2 e^2 f+15 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{315 b^2 d^3 f^5 \sqrt {c+d x}}+\frac {\left (\left (8 a^4 C d^4 f^4+a^3 b d^3 f^3 (11 C d e-7 c C f-18 B d f)-3 a^2 b^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )-a b^3 d f \left (C \left (184 d^3 e^3-66 c d^2 e^2 f-36 c^2 d e f^2-32 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d e f-19 c^2 f^2\right )\right )\right )+b^4 \left (C \left (128 d^4 e^4-40 c d^3 e^3 f-21 c^2 d^2 e^2 f^2-16 c^3 d e f^3-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 e^2 f-9 c^2 d e f^2-8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{315 b^2 d^3 f^5 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{945 b^2 d^3 f^4}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {2 \sqrt {-b c+a d} \left (8 a^4 C d^4 f^4+a^3 b d^3 f^3 (11 C d e-7 c C f-18 B d f)-3 a^2 b^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )-a b^3 d f \left (2 C \left (92 d^3 e^3-33 c d^2 e^2 f-18 c^2 d e f^2-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d e f-19 c^2 f^2\right )\right )\right )+b^4 \left (C \left (128 d^4 e^4-40 c d^3 e^3 f-21 c^2 d^2 e^2 f^2-16 c^3 d e f^3-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 e^2 f-9 c^2 d e f^2-8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {\left ((b e-a f) (d e-c f) \left (4 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (3 C d e-c C f-3 B d f)-3 a b^2 d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d e f+c^2 f^2\right )\right )-b^3 \left (C \left (128 d^3 e^3+24 c d^2 e^2 f+15 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{315 b^2 d^3 f^5 \sqrt {c+d x} \sqrt {e+f x}}\\ &=-\frac {2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac {a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{945 b^2 d^3 f^4}-\frac {2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{315 b d^3 f^3}-\frac {2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{63 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} (c+d x)^{3/2} \sqrt {e+f x}}{9 b d f}+\frac {2 \sqrt {-b c+a d} \left (8 a^4 C d^4 f^4+a^3 b d^3 f^3 (11 C d e-7 c C f-18 B d f)-3 a^2 b^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )-a b^3 d f \left (2 C \left (92 d^3 e^3-33 c d^2 e^2 f-18 c^2 d e f^2-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d e f-19 c^2 f^2\right )\right )\right )+b^4 \left (C \left (128 d^4 e^4-40 c d^3 e^3 f-21 c^2 d^2 e^2 f^2-16 c^3 d e f^3-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 e^2 f-9 c^2 d e f^2-8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {-b c+a d} (b e-a f) (d e-c f) \left (4 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (3 C d e-c C f-3 B d f)-3 a b^2 d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d e f+c^2 f^2\right )\right )-b^3 \left (C \left (128 d^3 e^3+24 c d^2 e^2 f+15 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^3 d^{7/2} f^5 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

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Mathematica [C]  time = 17.63, size = 12483, normalized size = 10.11 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]

[Out]

Result too large to show

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fricas [F]  time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C b x^{3} + {\left (C a + B b\right )} x^{2} + A a + {\left (B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{\sqrt {f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b*x^3 + (C*a + B*b)*x^2 + A*a + (B*a + A*b)*x)*sqrt(b*x + a)*sqrt(d*x + c)/sqrt(f*x + e), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{\sqrt {f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)*sqrt(d*x + c)/sqrt(f*x + e), x)

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maple [B]  time = 0.07, size = 15855, normalized size = 12.84 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{\sqrt {f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)*sqrt(d*x + c)/sqrt(f*x + e), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(e + f*x)^(1/2),x)

[Out]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(e + f*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\sqrt {e + f x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(C*x**2+B*x+A)*(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral((a + b*x)**(3/2)*sqrt(c + d*x)*(A + B*x + C*x**2)/sqrt(e + f*x), x)

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