3.1276 \(\int \frac {A+B x}{(d+e x)^{3/2} (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=415 \[ -\frac {2 e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-2 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 (-e) (B d-2 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)} \]

[Out]

-2*(A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2*(2*A*c^2*d^2-
b^2*e*(-2*A*e+B*d)-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)^(3/2)/d^2/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2*(A*b*e-2*A*c*d+B*b
*d)*EllipticF(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)*(1+e*x/d)^(1/2)/(-b)
^(3/2)/d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2*e*(2*A*c^2*d^2-b^2*e*(-2*A*e+B*d)-b*c*d*(2*A*e+B*d))*(c*
x^2+b*x)^(1/2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)

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Rubi [A]  time = 0.54, antiderivative size = 415, 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 = {822, 834, 843, 715, 112, 110, 117, 116} \[ -\frac {2 e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-2 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 (-e) (B d-2 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]) - (
2*e*(2*A*c^2*d^2 - b^2*e*(B*d - 2*A*e) - b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(b^2*d^2*(c*d - b*e)^2*Sqrt[d
+ e*x]) + (2*Sqrt[c]*(2*A*c^2*d^2 - b^2*e*(B*d - 2*A*e) - 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)])/((-b)^(3/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (
e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(b*B*d - 2*A*c*d + A*b*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)])/((-b)^(3/2)*d*(c*d - b*e)*Sqrt[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 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 834

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

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Mathematica [C]  time = 1.81, size = 367, normalized size = 0.88 \[ \frac {2 \left (i c e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^2 e (2 A e-B d)-b c d (2 A e+B d)+2 A c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+(b+c x) (d+e x) \left (b^2 e (2 A e-B d)-b c d (2 A e+B d)+2 A c^2 d^2\right )+b^2 e^2 x (b+c x) (B d-A e)+c^2 d^2 x (d+e x) (b B-A c)-i c e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) (-2 A b e+A c d+b B d) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-A (b+c x) (d+e x) (c d-b e)^2\right )}{b^2 d^2 \sqrt {x (b+c x)} \sqrt {d+e x} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2*e^2*(B*d - A*e)*x*(b + c*x) + c^2*(b*B - A*c)*d^2*x*(d + e*x) - A*(c*d - b*e)^2*(b + c*x)*(d + e*x) +
(2*A*c^2*d^2 + b^2*e*(-(B*d) + 2*A*e) - b*c*d*(B*d + 2*A*e))*(b + c*x)*(d + e*x) + I*Sqrt[b/c]*c*e*(2*A*c^2*d^
2 + b^2*e*(-(B*d) + 2*A*e) - b*c*d*(B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcS
inh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(c*d - b*e)*(b*B*d + A*c*d - 2*A*b*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)]))/(b^2*d^2*(c*d - b*e)^2*Sqrt[
x*(b + c*x)]*Sqrt[d + e*x])

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c^{2} e^{2} x^{6} + b^{2} d^{2} x^{2} + 2 \, {\left (c^{2} d e + b c e^{2}\right )} x^{5} + {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c d^{2} + b^{2} d e\right )} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
1.38Unable to transpose Error: Bad Argument Value

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maple [B]  time = 0.12, size = 1079, normalized size = 2.60 \[ -\frac {2 \left (2 A \,b^{2} c^{2} e^{3} x^{2}-2 A b \,c^{3} d \,e^{2} x^{2}+2 A \,c^{4} d^{2} e \,x^{2}-B \,b^{2} c^{2} d \,e^{2} x^{2}-B b \,c^{3} d^{2} e \,x^{2}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{4} e^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{3} c d \,e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{3} c d \,e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+2 A \,b^{3} c \,e^{3} x +4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{2} c^{2} d^{2} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{2} c^{2} d^{2} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-A \,b^{2} c^{2} d \,e^{2} x -2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A b \,c^{3} d^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A b \,c^{3} d^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-A b \,c^{3} d^{2} e x +2 A \,c^{4} d^{3} x -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{4} d \,e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{3} c \,d^{2} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-B \,b^{3} c d \,e^{2} x +\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{2} c^{2} d^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{2} c^{2} d^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-B b \,c^{3} d^{3} x +A \,b^{3} c d \,e^{2}-2 A \,b^{2} c^{2} d^{2} e +A b \,c^{3} d^{3}\right ) \sqrt {\left (c x +b \right ) x}}{\left (c x +b \right ) \left (b e -c d \right )^{2} \sqrt {e x +d}\, b^{2} c \,d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/x*(-A*x*b^2*c^2*d*e^2-B*x*b*c^3*d^3+A*b*c^3*d^3+2*A*x*c^4*d^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))*b^3*c*d^2*e+4*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))*b^2*c^2*d^2*e
+2*A*x^2*b^2*c^2*e^3+2*A*x^2*c^4*d^2*e+2*A*x*b^3*c*e^3-2*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))*b*c^3*d^3+2*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))*b*c^3*d^3-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))*b^4*
d*e^2+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))*b^2*c^2*d^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))*b^2*c^2*d^3-A*x*b*c^3*d^2*e-2*A*x^2*b*c^3*d*e^2-B*x^2*b^2*c^2*d*e^2-B*x^2*
b*c^3*d^2*e-B*x*b^3*c*d*e^2+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))*b^3*c*d*e^2-3*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))*b^2*c^2*d^2*e-4*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))*b^3*c*d*e^2-2*A*b^2*c^2*d
^2*e+A*b^3*c*d*e^2+2*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))*b^4*e^3)*((c*x+b)*x)^(1/2)/(c*x+b)/(b*e-c*d)^2/c/b^2/d^2/(e*x+d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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