3.4.88 \(\int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx\) [388]

Optimal. Leaf size=246 \[ \frac {2 \sqrt {d+e x} \sqrt {b x+c x^2}}{3 e}-\frac {2 \sqrt {-b} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(-b*e+2*c*d)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*
x+d)^(1/2)/e^2/c^(1/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+4/3*d*(-b*e+c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2)
,(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/e^2/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/
2)+2/3*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/e

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Rubi [A]
time = 0.13, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {748, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^2 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*e) - (2*Sqrt[-b]*(2*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*Sqrt[c]*e^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]
) + (4*Sqrt[-b]*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/S
qrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 113

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 117

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

Rule 118

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 729

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 748

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 857

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

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Mathematica [C] Result contains complex when optimal does not.
time = 6.63, size = 226, normalized size = 0.92 \begin {gather*} \frac {2 \left ((b+c x) (d+e x) (-2 c d+b e+c e x)+i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e (-c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )}{3 c e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/Sqrt[d + e*x],x]

[Out]

(2*((b + c*x)*(d + e*x)*(-2*c*d + b*e + c*e*x) + I*Sqrt[b/c]*c*e*(-2*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*Sqrt[b/c]*c*e*(-(c*d) + 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)]))/(3*c*e^2*Sqrt[x*(b +
c*x)]*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(466\) vs. \(2(198)=396\).
time = 0.45, size = 467, normalized size = 1.90

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 e}-\frac {2 b^{2} d \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{3 e c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (b -\frac {2 \left (b e +c d \right )}{3 e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(380\)
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \left (2 b^{2} d \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) e c -2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{2}-3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e +2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}-c^{3} e^{2} x^{3}-b \,c^{2} e^{2} x^{2}-c^{3} d e \,x^{2}-b \,c^{2} d e x \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) e^{2} c^{2}}\)	\(467\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.35, size = 341, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {x e + d} c^{2} e^{2} + {\left (2 \, c^{2} d^{2} - 2 \, b c d e - b^{2} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right )\right )} e^{\left (-3\right )}}{9 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*sqrt(c*x^2 + b*x)*sqrt(x*e + d)*c^2*e^2 + (2*c^2*d^2 - 2*b*c*d*e - b^2*e^2)*sqrt(c)*e^(1/2)*weierstrass
PInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^
3*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) + 3*(2*c^2*d*e - b*c*e^2)*sqrt(c)*e^(1/2)*weierstrassZe
ta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)
*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*
e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)))*e^(-3)/c^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{\sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*x))/sqrt(d + e*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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