3.408 \(\int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 241, 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 = {742, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + (4*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*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2])
- (2*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])/Sqr
t[-b]], (b*e)/(c*d)])/(3*c^(3/2)*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 742

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

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

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Mathematica [C]  time = 1.01, size = 246, normalized size = 1.02 \[ \frac {2 i c x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (2 b^2 e^2-5 b c d e+3 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-4 i b c e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 b (b+c x) (d+e x) (2 b e-c (4 d+e x))}{3 b c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.07, size = 460, normalized size = 1.91 \[ \frac {2 \sqrt {e x +d}\, \sqrt {\left (c x +b \right ) x}\, \left (c^{3} e^{2} x^{3}+b \,c^{2} e^{2} x^{2}+c^{3} d e \,x^{2}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c d e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c d e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+b \,c^{2} d e x \right )}{3 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(c*x^2+b*x)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(b*x + c*x^2)^(1/2),x)

[Out]

int((d + e*x)^(3/2)/(b*x + c*x^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(c*x**2+b*x)**(1/2),x)

[Out]

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

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