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

Optimal. Leaf size=371 \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2+2 b d f x (2 a C d f-b (6 B d f-5 C (c f+d e)))-6 a b d f (4 B d f-3 C (c f+d e))-\left (b^2 \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )\right )\right )}{24 b d^3 f^3}+\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (2 a d f \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-b \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )\right )}{8 d^{7/2} f^{7/2}}+\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f} \]

[Out]

1/8*(2*a*d*f*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2*A*d*f-B*(c*f+d*e)))-b*(C*(5*c^3*f^3+3*c^2*d*e*f^2+3*c
*d^2*e^2*f+5*d^3*e^3)+2*d*f*(4*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2))))*arctanh(f^(1/2)*(d*x+c)^(1
/2)/d^(1/2)/(f*x+e)^(1/2))/d^(7/2)/f^(7/2)+1/3*C*(b*x+a)^2*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d/f-1/24*(8*a^2*C*d^2
*f^2-6*a*b*d*f*(4*B*d*f-3*C*(c*f+d*e))-b^2*(C*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2)+6*d*f*(4*A*d*f-3*B*(c*f+d*e))
)+2*b*d*f*(2*a*C*d*f-b*(6*B*d*f-5*C*(c*f+d*e)))*x)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^3/f^3

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Rubi [A]  time = 0.51, antiderivative size = 369, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1615, 147, 63, 217, 206} \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2-2 b d f x (-2 a C d f+6 b B d f-5 b C (c f+d e))-6 a b d f (4 B d f-3 C (c f+d e))+b^2 \left (-\left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )\right )\right )}{24 b d^3 f^3}+\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (2 a d f \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-b \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )\right )}{8 d^{7/2} f^{7/2}}+\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d*f) - (Sqrt[c + d*x]*Sqrt[e + f*x]*(8*a^2*C*d^2*f^2 - 6*a*b*
d*f*(4*B*d*f - 3*C*(d*e + c*f)) - b^2*(C*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2) + 6*d*f*(4*A*d*f - 3*B*(d*e +
c*f))) - 2*b*d*f*(6*b*B*d*f - 2*a*C*d*f - 5*b*C*(d*e + c*f))*x))/(24*b*d^3*f^3) + ((2*a*d*f*(C*(3*d^2*e^2 + 2*
c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*
c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x
])/(Sqrt[d]*Sqrt[e + f*x])])/(8*d^(7/2)*f^(7/2))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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) \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx &=\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}+\frac {\int \frac {(a+b x) \left (-\frac {1}{2} b (4 b c C e+a C d e+a c C f-6 A b d f)+\frac {1}{2} b (6 b B d f-2 a C d f-5 b C (d e+c f)) x\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 b^2 d f}\\ &=\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2-6 a b d f (4 B d f-3 C (d e+c f))-b^2 \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )-2 b d f (6 b B d f-2 a C d f-5 b C (d e+c f)) x\right )}{24 b d^3 f^3}+\frac {\left (2 a d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 d^3 f^3}\\ &=\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2-6 a b d f (4 B d f-3 C (d e+c f))-b^2 \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )-2 b d f (6 b B d f-2 a C d f-5 b C (d e+c f)) x\right )}{24 b d^3 f^3}+\frac {\left (2 a d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{8 d^4 f^3}\\ &=\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2-6 a b d f (4 B d f-3 C (d e+c f))-b^2 \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )-2 b d f (6 b B d f-2 a C d f-5 b C (d e+c f)) x\right )}{24 b d^3 f^3}+\frac {\left (2 a d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{8 d^4 f^3}\\ &=\frac {C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d^2 f^2-6 a b d f (4 B d f-3 C (d e+c f))-b^2 \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )-2 b d f (6 b B d f-2 a C d f-5 b C (d e+c f)) x\right )}{24 b d^3 f^3}+\frac {\left (2 a d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{8 d^{7/2} f^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 1.96, size = 379, normalized size = 1.02 \[ \frac {\sqrt {e+f x} \left (3 \sqrt {d e-c f} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right ) \left (b \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )-2 a d f \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )\right )-\frac {d \sqrt {f} \sqrt {c+d x} (e+f x) \left (6 a d f (4 B d f+C (-3 c f-3 d e+2 d f x))+b \left (6 d f (4 A d f+B (-3 c f-3 d e+2 d f x))+C \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )}{\sqrt {\frac {d (e+f x)}{d e-c f}}}\right )}{24 d^3 f^{7/2} (c f-d e) \sqrt {\frac {d (e+f x)}{d e-c f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[e + f*x]*(-((d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(6*a*d*f*(4*B*d*f + C*(-3*d*e - 3*c*f + 2*d*f*x)) + b*(6*
d*f*(4*A*d*f + B*(-3*d*e - 3*c*f + 2*d*f*x)) + C*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10*e*f*x
+ 8*f^2*x^2)))))/Sqrt[(d*(e + f*x))/(d*e - c*f)]) + 3*Sqrt[d*e - c*f]*(-2*a*d*f*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*
c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) + b*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) + 2
*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))))*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e
- c*f]]))/(24*d^3*f^(7/2)*(-(d*e) + c*f)*Sqrt[(d*(e + f*x))/(d*e - c*f)])

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fricas [A]  time = 1.59, size = 720, normalized size = 1.94 \[ \left [-\frac {3 \, {\left (5 \, C b d^{3} e^{3} + 3 \, {\left (C b c d^{2} - 2 \, {\left (C a + B b\right )} d^{3}\right )} e^{2} f + {\left (3 \, C b c^{2} d - 4 \, {\left (C a + B b\right )} c d^{2} + 8 \, {\left (B a + A b\right )} d^{3}\right )} e f^{2} + {\left (5 \, C b c^{3} - 16 \, A a d^{3} - 6 \, {\left (C a + B b\right )} c^{2} d + 8 \, {\left (B a + A b\right )} c d^{2}\right )} f^{3}\right )} \sqrt {d f} \log \left (8 \, d^{2} f^{2} x^{2} + d^{2} e^{2} + 6 \, c d e f + c^{2} f^{2} + 4 \, {\left (2 \, d f x + d e + c f\right )} \sqrt {d f} \sqrt {d x + c} \sqrt {f x + e} + 8 \, {\left (d^{2} e f + c d f^{2}\right )} x\right ) - 4 \, {\left (8 \, C b d^{3} f^{3} x^{2} + 15 \, C b d^{3} e^{2} f + 2 \, {\left (7 \, C b c d^{2} - 9 \, {\left (C a + B b\right )} d^{3}\right )} e f^{2} + 3 \, {\left (5 \, C b c^{2} d - 6 \, {\left (C a + B b\right )} c d^{2} + 8 \, {\left (B a + A b\right )} d^{3}\right )} f^{3} - 2 \, {\left (5 \, C b d^{3} e f^{2} + {\left (5 \, C b c d^{2} - 6 \, {\left (C a + B b\right )} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {f x + e}}{96 \, d^{4} f^{4}}, \frac {3 \, {\left (5 \, C b d^{3} e^{3} + 3 \, {\left (C b c d^{2} - 2 \, {\left (C a + B b\right )} d^{3}\right )} e^{2} f + {\left (3 \, C b c^{2} d - 4 \, {\left (C a + B b\right )} c d^{2} + 8 \, {\left (B a + A b\right )} d^{3}\right )} e f^{2} + {\left (5 \, C b c^{3} - 16 \, A a d^{3} - 6 \, {\left (C a + B b\right )} c^{2} d + 8 \, {\left (B a + A b\right )} c d^{2}\right )} f^{3}\right )} \sqrt {-d f} \arctan \left (\frac {{\left (2 \, d f x + d e + c f\right )} \sqrt {-d f} \sqrt {d x + c} \sqrt {f x + e}}{2 \, {\left (d^{2} f^{2} x^{2} + c d e f + {\left (d^{2} e f + c d f^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, C b d^{3} f^{3} x^{2} + 15 \, C b d^{3} e^{2} f + 2 \, {\left (7 \, C b c d^{2} - 9 \, {\left (C a + B b\right )} d^{3}\right )} e f^{2} + 3 \, {\left (5 \, C b c^{2} d - 6 \, {\left (C a + B b\right )} c d^{2} + 8 \, {\left (B a + A b\right )} d^{3}\right )} f^{3} - 2 \, {\left (5 \, C b d^{3} e f^{2} + {\left (5 \, C b c d^{2} - 6 \, {\left (C a + B b\right )} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {f x + e}}{48 \, d^{4} f^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(3*(5*C*b*d^3*e^3 + 3*(C*b*c*d^2 - 2*(C*a + B*b)*d^3)*e^2*f + (3*C*b*c^2*d - 4*(C*a + B*b)*c*d^2 + 8*(B
*a + A*b)*d^3)*e*f^2 + (5*C*b*c^3 - 16*A*a*d^3 - 6*(C*a + B*b)*c^2*d + 8*(B*a + A*b)*c*d^2)*f^3)*sqrt(d*f)*log
(8*d^2*f^2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 4*(2*d*f*x + d*e + c*f)*sqrt(d*f)*sqrt(d*x + c)*sqrt(f*x + e)
+ 8*(d^2*e*f + c*d*f^2)*x) - 4*(8*C*b*d^3*f^3*x^2 + 15*C*b*d^3*e^2*f + 2*(7*C*b*c*d^2 - 9*(C*a + B*b)*d^3)*e*
f^2 + 3*(5*C*b*c^2*d - 6*(C*a + B*b)*c*d^2 + 8*(B*a + A*b)*d^3)*f^3 - 2*(5*C*b*d^3*e*f^2 + (5*C*b*c*d^2 - 6*(C
*a + B*b)*d^3)*f^3)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^4*f^4), 1/48*(3*(5*C*b*d^3*e^3 + 3*(C*b*c*d^2 - 2*(C*a
+ B*b)*d^3)*e^2*f + (3*C*b*c^2*d - 4*(C*a + B*b)*c*d^2 + 8*(B*a + A*b)*d^3)*e*f^2 + (5*C*b*c^3 - 16*A*a*d^3 -
6*(C*a + B*b)*c^2*d + 8*(B*a + A*b)*c*d^2)*f^3)*sqrt(-d*f)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)*sqrt(d*
x + c)*sqrt(f*x + e)/(d^2*f^2*x^2 + c*d*e*f + (d^2*e*f + c*d*f^2)*x)) + 2*(8*C*b*d^3*f^3*x^2 + 15*C*b*d^3*e^2*
f + 2*(7*C*b*c*d^2 - 9*(C*a + B*b)*d^3)*e*f^2 + 3*(5*C*b*c^2*d - 6*(C*a + B*b)*c*d^2 + 8*(B*a + A*b)*d^3)*f^3
- 2*(5*C*b*d^3*e*f^2 + (5*C*b*c*d^2 - 6*(C*a + B*b)*d^3)*f^3)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^4*f^4)]

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giac [A]  time = 1.97, size = 447, normalized size = 1.20 \[ \frac {{\left (\sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt {d x + c} {\left (2 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )} C b}{d^{4} f} - \frac {13 \, C b c d^{11} f^{4} - 6 \, C a d^{12} f^{4} - 6 \, B b d^{12} f^{4} + 5 \, C b d^{12} f^{3} e}{d^{15} f^{5}}\right )} + \frac {3 \, {\left (11 \, C b c^{2} d^{11} f^{4} - 10 \, C a c d^{12} f^{4} - 10 \, B b c d^{12} f^{4} + 8 \, B a d^{13} f^{4} + 8 \, A b d^{13} f^{4} + 8 \, C b c d^{12} f^{3} e - 6 \, C a d^{13} f^{3} e - 6 \, B b d^{13} f^{3} e + 5 \, C b d^{13} f^{2} e^{2}\right )}}{d^{15} f^{5}}\right )} + \frac {3 \, {\left (5 \, C b c^{3} f^{3} - 6 \, C a c^{2} d f^{3} - 6 \, B b c^{2} d f^{3} + 8 \, B a c d^{2} f^{3} + 8 \, A b c d^{2} f^{3} - 16 \, A a d^{3} f^{3} + 3 \, C b c^{2} d f^{2} e - 4 \, C a c d^{2} f^{2} e - 4 \, B b c d^{2} f^{2} e + 8 \, B a d^{3} f^{2} e + 8 \, A b d^{3} f^{2} e + 3 \, C b c d^{2} f e^{2} - 6 \, C a d^{3} f e^{2} - 6 \, B b d^{3} f e^{2} + 5 \, C b d^{3} e^{3}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt {d f} d^{3} f^{3}}\right )} d}{24 \, {\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)*C*b/(d^4*f) - (13*C*b*c*d^11
*f^4 - 6*C*a*d^12*f^4 - 6*B*b*d^12*f^4 + 5*C*b*d^12*f^3*e)/(d^15*f^5)) + 3*(11*C*b*c^2*d^11*f^4 - 10*C*a*c*d^1
2*f^4 - 10*B*b*c*d^12*f^4 + 8*B*a*d^13*f^4 + 8*A*b*d^13*f^4 + 8*C*b*c*d^12*f^3*e - 6*C*a*d^13*f^3*e - 6*B*b*d^
13*f^3*e + 5*C*b*d^13*f^2*e^2)/(d^15*f^5)) + 3*(5*C*b*c^3*f^3 - 6*C*a*c^2*d*f^3 - 6*B*b*c^2*d*f^3 + 8*B*a*c*d^
2*f^3 + 8*A*b*c*d^2*f^3 - 16*A*a*d^3*f^3 + 3*C*b*c^2*d*f^2*e - 4*C*a*c*d^2*f^2*e - 4*B*b*c*d^2*f^2*e + 8*B*a*d
^3*f^2*e + 8*A*b*d^3*f^2*e + 3*C*b*c*d^2*f*e^2 - 6*C*a*d^3*f*e^2 - 6*B*b*d^3*f*e^2 + 5*C*b*d^3*e^3)*log(abs(-s
qrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^3*f^3))*d/abs(d)

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maple [B]  time = 0.03, size = 1199, normalized size = 3.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(18*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*d^3*e^2*f+48*A*(d*f)^
(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*d^2*f^2+48*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*d^2*f^2+30*C*(d*f)^(1/2)*((
d*x+c)*(f*x+e))^(1/2)*b*c^2*f^2+30*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*d^2*e^2-24*A*ln(1/2*(2*d*f*x+c*f+d*
e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*c*d^2*f^3-24*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x
+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*d^3*e*f^2-24*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^
(1/2))/(d*f)^(1/2))*a*c*d^2*f^3-24*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2
))*a*d^3*e*f^2+18*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*c^2*d*f^3+18
*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*d^3*e^2*f+18*C*ln(1/2*(2*d*f*
x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*c^2*d*f^3+48*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x
+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*d^3*f^3+16*C*x^2*b*d^2*f^2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+
12*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*c*d^2*e*f^2+12*C*ln(1/2*(2*
d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*c*d^2*e*f^2-9*C*ln(1/2*(2*d*f*x+c*f+d*e+2*
((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*c^2*d*e*f^2-9*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e)
)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*c*d^2*e^2*f+24*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*d^2*f^2+24*C*(d*f
)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*d^2*f^2-36*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c*d*f^2-36*B*(d*f)^(1/2
)*((d*x+c)*(f*x+e))^(1/2)*b*d^2*e*f-36*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*c*d*f^2-36*C*(d*f)^(1/2)*((d*x+
c)*(f*x+e))^(1/2)*a*d^2*e*f-15*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b
*c^3*f^3-15*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b*d^3*e^3+28*C*(d*f)
^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c*d*e*f-20*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*d^2*e*f-20*C*(d*f)^(1/2)
*((d*x+c)*(f*x+e))^(1/2)*x*b*c*d*f^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/f^3/d^3/(d*f)^(1/2)/((d*x+c)*(f*x+e))^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for
more details)Is c*f-d*e zero or nonzero?

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mupad [B]  time = 105.19, size = 2621, normalized size = 7.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((((c + d*x)^(1/2) - c^(1/2))*(2*A*b*c*f + 2*A*b*d*e))/(f^3*((e + f*x)^(1/2) - e^(1/2))) + (((c + d*x)^(1/2) -
c^(1/2))^3*(2*A*b*c*f + 2*A*b*d*e))/(d*f^2*((e + f*x)^(1/2) - e^(1/2))^3) - (8*A*b*c^(1/2)*e^(1/2)*((c + d*x)
^(1/2) - c^(1/2))^2)/(f^2*((e + f*x)^(1/2) - e^(1/2))^2))/(((c + d*x)^(1/2) - c^(1/2))^4/((e + f*x)^(1/2) - e^
(1/2))^4 + d^2/f^2 - (2*d*((c + d*x)^(1/2) - c^(1/2))^2)/(f*((e + f*x)^(1/2) - e^(1/2))^2)) - ((((c + d*x)^(1/
2) - c^(1/2))*((3*C*a*d^3*e^2)/2 + (3*C*a*c^2*d*f^2)/2 + C*a*c*d^2*e*f))/(f^6*((e + f*x)^(1/2) - e^(1/2))) - (
((c + d*x)^(1/2) - c^(1/2))^3*((11*C*a*c^2*f^2)/2 + (11*C*a*d^2*e^2)/2 + 25*C*a*c*d*e*f))/(f^5*((e + f*x)^(1/2
) - e^(1/2))^3) + (((c + d*x)^(1/2) - c^(1/2))^7*((3*C*a*c^2*f^2)/2 + (3*C*a*d^2*e^2)/2 + C*a*c*d*e*f))/(d^2*f
^3*((e + f*x)^(1/2) - e^(1/2))^7) - (((c + d*x)^(1/2) - c^(1/2))^5*((11*C*a*c^2*f^2)/2 + (11*C*a*d^2*e^2)/2 +
25*C*a*c*d*e*f))/(d*f^4*((e + f*x)^(1/2) - e^(1/2))^5) + (c^(1/2)*e^(1/2)*((c + d*x)^(1/2) - c^(1/2))^4*(32*C*
a*c*f + 32*C*a*d*e))/(f^4*((e + f*x)^(1/2) - e^(1/2))^4))/(((c + d*x)^(1/2) - c^(1/2))^8/((e + f*x)^(1/2) - e^
(1/2))^8 + d^4/f^4 - (4*d*((c + d*x)^(1/2) - c^(1/2))^6)/(f*((e + f*x)^(1/2) - e^(1/2))^6) - (4*d^3*((c + d*x)
^(1/2) - c^(1/2))^2)/(f^3*((e + f*x)^(1/2) - e^(1/2))^2) + (6*d^2*((c + d*x)^(1/2) - c^(1/2))^4)/(f^2*((e + f*
x)^(1/2) - e^(1/2))^4)) - ((((c + d*x)^(1/2) - c^(1/2))^3*((85*C*b*d^4*e^3)/12 + (85*C*b*c^3*d*f^3)/12 + (17*C
*b*c*d^3*e^2*f)/4 + (17*C*b*c^2*d^2*e*f^2)/4))/(f^8*((e + f*x)^(1/2) - e^(1/2))^3) - (((c + d*x)^(1/2) - c^(1/
2))*((5*C*b*d^5*e^3)/4 + (5*C*b*c^3*d^2*f^3)/4 + (3*C*b*c*d^4*e^2*f)/4 + (3*C*b*c^2*d^3*e*f^2)/4))/(f^9*((e +
f*x)^(1/2) - e^(1/2))) - (((c + d*x)^(1/2) - c^(1/2))^5*((33*C*b*c^3*f^3)/2 + (33*C*b*d^3*e^3)/2 + (327*C*b*c*
d^2*e^2*f)/2 + (327*C*b*c^2*d*e*f^2)/2))/(f^7*((e + f*x)^(1/2) - e^(1/2))^5) - (((c + d*x)^(1/2) - c^(1/2))^11
*((5*C*b*c^3*f^3)/4 + (5*C*b*d^3*e^3)/4 + (3*C*b*c*d^2*e^2*f)/4 + (3*C*b*c^2*d*e*f^2)/4))/(d^3*f^4*((e + f*x)^
(1/2) - e^(1/2))^11) + (((c + d*x)^(1/2) - c^(1/2))^9*((85*C*b*c^3*f^3)/12 + (85*C*b*d^3*e^3)/12 + (17*C*b*c*d
^2*e^2*f)/4 + (17*C*b*c^2*d*e*f^2)/4))/(d^2*f^5*((e + f*x)^(1/2) - e^(1/2))^9) - (((c + d*x)^(1/2) - c^(1/2))^
7*((33*C*b*c^3*f^3)/2 + (33*C*b*d^3*e^3)/2 + (327*C*b*c*d^2*e^2*f)/2 + (327*C*b*c^2*d*e*f^2)/2))/(d*f^6*((e +
f*x)^(1/2) - e^(1/2))^7) + (c^(1/2)*e^(1/2)*((c + d*x)^(1/2) - c^(1/2))^6*(128*C*b*c^2*f^2 + 128*C*b*d^2*e^2 +
(896*C*b*c*d*e*f)/3))/(f^6*((e + f*x)^(1/2) - e^(1/2))^6) + (64*C*b*c^(3/2)*e^(3/2)*((c + d*x)^(1/2) - c^(1/2
))^8)/(f^4*((e + f*x)^(1/2) - e^(1/2))^8) + (64*C*b*c^(3/2)*d^2*e^(3/2)*((c + d*x)^(1/2) - c^(1/2))^4)/(f^6*((
e + f*x)^(1/2) - e^(1/2))^4))/(((c + d*x)^(1/2) - c^(1/2))^12/((e + f*x)^(1/2) - e^(1/2))^12 + d^6/f^6 - (6*d*
((c + d*x)^(1/2) - c^(1/2))^10)/(f*((e + f*x)^(1/2) - e^(1/2))^10) - (6*d^5*((c + d*x)^(1/2) - c^(1/2))^2)/(f^
5*((e + f*x)^(1/2) - e^(1/2))^2) + (15*d^4*((c + d*x)^(1/2) - c^(1/2))^4)/(f^4*((e + f*x)^(1/2) - e^(1/2))^4)
- (20*d^3*((c + d*x)^(1/2) - c^(1/2))^6)/(f^3*((e + f*x)^(1/2) - e^(1/2))^6) + (15*d^2*((c + d*x)^(1/2) - c^(1
/2))^8)/(f^2*((e + f*x)^(1/2) - e^(1/2))^8)) - ((((c + d*x)^(1/2) - c^(1/2))*((3*B*b*d^3*e^2)/2 + (3*B*b*c^2*d
*f^2)/2 + B*b*c*d^2*e*f))/(f^6*((e + f*x)^(1/2) - e^(1/2))) - (((c + d*x)^(1/2) - c^(1/2))^3*((11*B*b*c^2*f^2)
/2 + (11*B*b*d^2*e^2)/2 + 25*B*b*c*d*e*f))/(f^5*((e + f*x)^(1/2) - e^(1/2))^3) + (((c + d*x)^(1/2) - c^(1/2))^
7*((3*B*b*c^2*f^2)/2 + (3*B*b*d^2*e^2)/2 + B*b*c*d*e*f))/(d^2*f^3*((e + f*x)^(1/2) - e^(1/2))^7) - (((c + d*x)
^(1/2) - c^(1/2))^5*((11*B*b*c^2*f^2)/2 + (11*B*b*d^2*e^2)/2 + 25*B*b*c*d*e*f))/(d*f^4*((e + f*x)^(1/2) - e^(1
/2))^5) + (c^(1/2)*e^(1/2)*((c + d*x)^(1/2) - c^(1/2))^4*(32*B*b*c*f + 32*B*b*d*e))/(f^4*((e + f*x)^(1/2) - e^
(1/2))^4))/(((c + d*x)^(1/2) - c^(1/2))^8/((e + f*x)^(1/2) - e^(1/2))^8 + d^4/f^4 - (4*d*((c + d*x)^(1/2) - c^
(1/2))^6)/(f*((e + f*x)^(1/2) - e^(1/2))^6) - (4*d^3*((c + d*x)^(1/2) - c^(1/2))^2)/(f^3*((e + f*x)^(1/2) - e^
(1/2))^2) + (6*d^2*((c + d*x)^(1/2) - c^(1/2))^4)/(f^2*((e + f*x)^(1/2) - e^(1/2))^4)) + ((((c + d*x)^(1/2) -
c^(1/2))*(2*B*a*c*f + 2*B*a*d*e))/(f^3*((e + f*x)^(1/2) - e^(1/2))) + (((c + d*x)^(1/2) - c^(1/2))^3*(2*B*a*c*
f + 2*B*a*d*e))/(d*f^2*((e + f*x)^(1/2) - e^(1/2))^3) - (8*B*a*c^(1/2)*e^(1/2)*((c + d*x)^(1/2) - c^(1/2))^2)/
(f^2*((e + f*x)^(1/2) - e^(1/2))^2))/(((c + d*x)^(1/2) - c^(1/2))^4/((e + f*x)^(1/2) - e^(1/2))^4 + d^2/f^2 -
(2*d*((c + d*x)^(1/2) - c^(1/2))^2)/(f*((e + f*x)^(1/2) - e^(1/2))^2)) - (4*A*a*atan((d*((e + f*x)^(1/2) - e^(
1/2)))/((-d*f)^(1/2)*((c + d*x)^(1/2) - c^(1/2)))))/(-d*f)^(1/2) + (B*b*atanh((f^(1/2)*((c + d*x)^(1/2) - c^(1
/2)))/(d^(1/2)*((e + f*x)^(1/2) - e^(1/2))))*(3*c^2*f^2 + 3*d^2*e^2 + 2*c*d*e*f))/(2*d^(5/2)*f^(5/2)) + (C*a*a
tanh((f^(1/2)*((c + d*x)^(1/2) - c^(1/2)))/(d^(1/2)*((e + f*x)^(1/2) - e^(1/2))))*(3*c^2*f^2 + 3*d^2*e^2 + 2*c
*d*e*f))/(2*d^(5/2)*f^(5/2)) - (2*A*b*atanh((f^(1/2)*((c + d*x)^(1/2) - c^(1/2)))/(d^(1/2)*((e + f*x)^(1/2) -
e^(1/2))))*(c*f + d*e))/(d^(3/2)*f^(3/2)) - (2*B*a*atanh((f^(1/2)*((c + d*x)^(1/2) - c^(1/2)))/(d^(1/2)*((e +
f*x)^(1/2) - e^(1/2))))*(c*f + d*e))/(d^(3/2)*f^(3/2)) - (C*b*atanh((f^(1/2)*((c + d*x)^(1/2) - c^(1/2)))/(d^(
1/2)*((e + f*x)^(1/2) - e^(1/2))))*(c*f + d*e)*(5*c^2*f^2 + 5*d^2*e^2 - 2*c*d*e*f))/(4*d^(7/2)*f^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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