3.1269 \(\int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx\)

Optimal. Leaf size=369 \[ -\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{3 d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d^2 e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d-A e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d e \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)} \]

[Out]

2/3*(2*A*e*(-b*e+2*c*d)-B*d*(b*e+c*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2
)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/d^2/e/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(-A*e+B*d)*El
lipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)
/d/e/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(-A*e+B*d)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^(3/2)-2/
3*(2*A*e*(-b*e+2*c*d)-B*d*(b*e+c*d))*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)

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Rubi [A]  time = 0.48, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {834, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{3 d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d^2 e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d-A e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d e \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*(2*A*e*(2*c*d - b*e) - B*d*(c*d + b*e
))*Sqrt[b*x + c*x^2])/(3*d^2*(c*d - b*e)^2*Sqrt[d + e*x]) + (2*Sqrt[-b]*Sqrt[c]*(2*A*e*(2*c*d - b*e) - B*d*(c*
d + b*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*d^2*e*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[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*d*e*(c*d - b*e)*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 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)^{5/2} \sqrt {b x+c x^2}} \, dx &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} (b B d-3 A c d+2 A b e)-\frac {1}{2} c (B d-A e) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{3 d (c d-b e)}\\ &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {1}{4} c d (2 b B d-3 A c d+A b e)+\frac {1}{4} c (2 A e (2 c d-b e)-B d (c d+b e)) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 d^2 (c d-b e)^2}\\ &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {(c (B d-A e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 d e (c d-b e)}+\frac {(c (2 A e (2 c d-b e)-B d (c d+b e))) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 d^2 e (c d-b e)^2}\\ &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {\left (c (B d-A e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 d e (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (c (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 d^2 e (c d-b e)^2 \sqrt {b x+c x^2}}\\ &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {\left (c (2 A e (2 c d-b e)-B d (c d+b e)) \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 d^2 e (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c (B d-A 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}{3 d e (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {c} (2 A e (2 c d-b e)-B d (c d+b e)) \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 d^2 e (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d-A 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 )}{3 d e (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 2.04, size = 347, normalized size = 0.94 \[ \frac {2 \left (b e x (b+c x) \left (A e (b e (3 d+2 e x)-c d (5 d+4 e x))+B d \left (b e^2 x+c d (2 d+e x)\right )\right )-c \sqrt {\frac {b}{c}} (d+e x) \left (-i e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) (3 A c d-b (2 A e+B d)) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (2 A e (b e-2 c d)+B d (b e+c d)) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) (2 A e (b e-2 c d)+B d (b e+c d))\right )\right )}{3 b d^2 e \sqrt {x (b+c x)} (d+e x)^{3/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*e*x*(b + c*x)*(B*d*(b*e^2*x + c*d*(2*d + e*x)) + A*e*(b*e*(3*d + 2*e*x) - c*d*(5*d + 4*e*x))) - Sqrt[b/c
]*c*(d + e*x)*(Sqrt[b/c]*(2*A*e*(-2*c*d + b*e) + B*d*(c*d + b*e))*(b + c*x)*(d + e*x) + I*b*e*(2*A*e*(-2*c*d +
b*e) + B*d*(c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (
c*d)/(b*e)] - I*e*(c*d - b*e)*(3*A*c*d - b*(B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elliptic
F[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b*d^2*e*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.12, size = 1635, normalized size = 4.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((c*x+b)*x)^(1/2)*(B*b*c^2*d*e^3*x^3+3*A*x*b^2*c*d*e^3+B*c^3*d^2*e^2*x^3+B*EllipticF(((c*x+b)/b)^(1/2),(1/
(b*e-c*d)*b*e)^(1/2))*b*c^2*d^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+2*A*EllipticE(
((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)
^(1/2)+2*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d
)*c)^(1/2)*(-1/b*c*x)^(1/2)+B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*d^2*e^2*((c*x+b)/b)^(1/
2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^2*
d^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+2*A*b^2*c*e^4*x^2-5*A*c^3*d^2*e^2*x^2+2*A*
b*c^2*e^4*x^3-4*A*c^3*d*e^3*x^3+2*B*c^3*d^3*e*x^2-A*b*c^2*d*e^3*x^2+B*b^2*c*d*e^3*x^2+B*b*c^2*d^2*e^2*x^2-5*A*
b*c^2*d^2*e^2*x+2*B*b*c^2*d^3*e*x-6*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c*d^2*e^2*((c*x
+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(
1/2))*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+A*EllipticF(((c*x+b)/b)^(1/2
),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-A*Ell
ipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(
-1/b*c*x)^(1/2)+B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)
/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c*d^3*e*((c*x+
b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-6*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1
/2))*x*b^2*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*A*EllipticE(((c*x+b)/b)^(
1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+
A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^
(1/2)*(-1/b*c*x)^(1/2)-A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^2*d^2*e^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^2
*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c
*d)*b*e)^(1/2))*x*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+B*EllipticF(((
c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*
x)^(1/2))/(c*x+b)/x/(b*e-c*d)^2/(e*x+d)^(3/2)/c/d^2/e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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