3.1281 \(\int \frac {(A+B x) \sqrt {d+e x}}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=420 \[ -\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-8 b c (A e+B d)+16 A c^2 d+3 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (A b e-8 A c d+4 b B d)-c x \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)} \]

[Out]

-2/3*(A*b-(-2*A*c+B*b)*x)*(e*x+d)^(1/2)/b^2/(c*x^2+b*x)^(3/2)-2/3*(b*(-b*e+c*d)*(A*b*e-8*A*c*d+4*B*b*d)-c*(16*
A*c^2*d^2+b^2*e*(A*e+7*B*d)-8*b*c*d*(2*A*e+B*d))*x)*(e*x+d)^(1/2)/b^4/d/(-b*e+c*d)/(c*x^2+b*x)^(1/2)-2/3*(16*A
*c^2*d^2+b^2*e*(A*e+7*B*d)-8*b*c*d*(2*A*e+B*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*
x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/d/(-b*e+c*d)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(16*A*c^2*
d+3*b^2*B*e-8*b*c*(A*e+B*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(c*x/b+1)^(1/2)*(1+
e*x/d)^(1/2)/(-b)^(7/2)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {820, 822, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \sqrt {d+e x} \left (b (c d-b e) (A b e-8 A c d+4 b B d)-c x \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-8 b c (A e+B d)+16 A c^2 d+3 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(3*b^2*(b*x + c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(4*b*
B*d - 8*A*c*d + A*b*e) - c*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*x))/(3*b^4*d*(c*d - b*
e)*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1
+ (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b
*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(16*A*c^2*d + 3*b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c
*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*Sqrt[c]*Sqr
t[d + e*x]*Sqrt[b*x + c*x^2])

Rule 110

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

Rule 112

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

Rule 116

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

Rule 117

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

Rule 715

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(Sqrt[x]*Sqrt[b + c*x])/Sqrt[
b*x + c*x^2], Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 820

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

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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 {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} (8 A c d-b (4 B d+A e))-\frac {3}{2} (b B-2 A c) e x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} b d e \left (4 b B c d-8 A c^2 d-3 b^2 B e+7 A b c e\right )-\frac {1}{4} c e \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}\\ &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}\\ &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}\\ &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 2.27, size = 441, normalized size = 1.05 \[ -\frac {2 \left (\sqrt {\frac {b}{c}} (d+e x) \left (c d x^2 (b+c x) \left (b c (7 A e+5 B d)-8 A c^2 d-4 b^2 B e\right )+b c d x^2 (b B-A c) (c d-b e)+x (b+c x)^2 (c d-b e) (A b e-8 A c d+3 b B d)+A b d (b+c x)^2 (c d-b e)\right )+x (b+c x) \left (i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) (8 A c d-b (A e+4 B d)) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{3 b^4 d \sqrt {\frac {b}{c}} (x (b+c x))^{3/2} \sqrt {d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(Sqrt[b/c]*(d + e*x)*(b*c*(b*B - A*c)*d*(c*d - b*e)*x^2 + c*d*(-8*A*c^2*d - 4*b^2*B*e + b*c*(5*B*d + 7*A*e
))*x^2*(b + c*x) + A*b*d*(c*d - b*e)*(b + c*x)^2 + (c*d - b*e)*(3*b*B*d - 8*A*c*d + A*b*e)*x*(b + c*x)^2) + x*
(b + c*x)*(Sqrt[b/c]*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*(b + c*x)*(d + e*x) + I*b*e*
(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip
ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c*d - b*(4*B*d + A*e))*Sqrt[1 + b/(c*
x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^4*Sqrt[b/c]*d*(c*d
- b*e)*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*(B*x + A)*sqrt(e*x + d)/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2), x)

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maple [B]  time = 0.13, size = 2503, normalized size = 5.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3/x^2*(A*b^4*c*d^2*e+8*B*x^2*b^3*c^2*d^2*e-3*B*x*b^3*c^2*d^3+A*x^2*b^4*c*e^3+16*A*x^4*c^5*d^2*e-8*B*x^3*b*c
^4*d^3-24*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*
e-c*d)*b*e)^(1/2))*x*b^3*c^2*d^2*e+11*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^2*d^2*e+24*A*x^2*b*c^4*d^3-12*B*x^2*b^2*c^3*d^3+A*x^4
*b^2*c^3*e^3+6*A*x*b^2*c^3*d^3+2*A*x^3*b^3*c^2*e^3+16*A*x^3*c^5*d^3-17*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)
*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^2*d*e^2+32*A*((c*x+b
)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*
x^2*b^2*c^3*d^2*e-17*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(
1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c*d*e^2-15*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/
2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c*d^2*e+16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*
d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^4*d^3+8*B*((c*x+b)/b
)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2
*b^2*c^3*d^3-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(
1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^3*d^3+A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^4*c*e^3-16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(
1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^4*d^3+16*A*((c*x+b)/b)^(1/2
)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^3
*d^3-16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-
c*d)*b*e)^(1/2))*x*b^2*c^3*d^3+7*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE((
(c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*d*e^2+8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b
*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^2*d^3-3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)
/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*d*e^2-8*B*((c*
x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2
))*x*b^3*c^2*d^3+7*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/
2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^4*c*d*e^2-15*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/
2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^2*d^2*e+8*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e
-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^2*d*e^2-24*A*((
c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1
/2))*x^2*b^2*c^3*d^2*e-3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/
b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^4*c*d*e^2+32*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*
x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^2*d^2*e+8*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/
(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c*d*e^2+11*B*((
c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1
/2))*x*b^4*c*d^2*e+A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/
2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*e^3-16*A*x^4*b*c^4*d*e^2+2*A*x*b^4*c*d*e^2-8*A*x*b^3*c^2*d^2*e-5*A*x^2*b^3*c
^2*d*e^2+3*B*x*b^4*c*d^2*e+3*B*x^2*b^4*c*d*e^2-5*B*x^3*b^2*c^3*d^2*e-19*A*x^2*b^2*c^3*d^2*e+11*B*x^3*b^3*c^2*d
*e^2-24*A*x^3*b^2*c^3*d*e^2+8*A*x^3*b*c^4*d^2*e+7*B*x^4*b^2*c^3*d*e^2-8*B*x^4*b*c^4*d^2*e-A*b^3*c^2*d^3)*((c*x
+b)*x)^(1/2)/b^4/c/d/(c*x+b)^2/(b*e-c*d)/(e*x+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(b*x + c*x^2)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(1/2))/(b*x + c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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