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

Optimal. Leaf size=99 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{4 c^{5/2}}-\frac {\sqrt {b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {779, 620, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{4 c^{5/2}}-\frac {\sqrt {b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*b*B*e - 4*c*(B*d + A*e) - 2*B*c*e*x)*Sqrt[b*x + c*x^2])/(4*c^2) + ((8*A*c^2*d + 3*b^2*B*e - 4*b*c*(B*d +
A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(5/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)}{\sqrt {b x+c x^2}} \, dx &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {b x+c x^2}}{4 c^2}+\frac {\left (\frac {3}{2} b^2 B e+2 c (2 A c d-b (B d+A e))\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{4 c^2}\\ &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {b x+c x^2}}{4 c^2}+\frac {\left (\frac {3}{2} b^2 B e+2 c (2 A c d-b (B d+A e))\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{2 c^2}\\ &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {b x+c x^2}}{4 c^2}+\frac {\left (8 A c^2 d+3 b^2 B e-4 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 115, normalized size = 1.16 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {\sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} (4 A c e+B (-3 b e+4 c d+2 c e x))\right )}{4 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(4*A*c*e + B*(4*c*d - 3*b*e + 2*c*e*x)) + ((8*A*c^2*d + 3*b^2*B*e - 4*b*c*(B*d + A
*e))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[x]*Sqrt[1 + (c*x)/b])))/(4*c^(5/2))

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IntegrateAlgebraic [A]  time = 0.54, size = 112, normalized size = 1.13 \begin {gather*} \frac {\log \left (-2 c^{5/2} \sqrt {b x+c x^2}+b c^2+2 c^3 x\right ) \left (4 A b c e-8 A c^2 d-3 b^2 B e+4 b B c d\right )}{8 c^{5/2}}+\frac {\sqrt {b x+c x^2} (4 A c e-3 b B e+4 B c d+2 B c e x)}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*B*c*d - 3*b*B*e + 4*A*c*e + 2*B*c*e*x)*Sqrt[b*x + c*x^2])/(4*c^2) + ((4*b*B*c*d - 8*A*c^2*d - 3*b^2*B*e +
4*A*b*c*e)*Log[b*c^2 + 2*c^3*x - 2*c^(5/2)*Sqrt[b*x + c*x^2]])/(8*c^(5/2))

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fricas [A]  time = 0.43, size = 217, normalized size = 2.19 \begin {gather*} \left [\frac {{\left (4 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (3 \, B b^{2} - 4 \, A b c\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, B c^{2} e x + 4 \, B c^{2} d - {\left (3 \, B b c - 4 \, A c^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{8 \, c^{3}}, \frac {{\left (4 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (3 \, B b^{2} - 4 \, A b c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (2 \, B c^{2} e x + 4 \, B c^{2} d - {\left (3 \, B b c - 4 \, A c^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{4 \, c^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((4*(B*b*c - 2*A*c^2)*d - (3*B*b^2 - 4*A*b*c)*e)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2
*(2*B*c^2*e*x + 4*B*c^2*d - (3*B*b*c - 4*A*c^2)*e)*sqrt(c*x^2 + b*x))/c^3, 1/4*((4*(B*b*c - 2*A*c^2)*d - (3*B*
b^2 - 4*A*b*c)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (2*B*c^2*e*x + 4*B*c^2*d - (3*B*b*c - 4*
A*c^2)*e)*sqrt(c*x^2 + b*x))/c^3]

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giac [A]  time = 0.28, size = 110, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (\frac {2 \, B x e}{c} + \frac {4 \, B c d - 3 \, B b e + 4 \, A c e}{c^{2}}\right )} + \frac {{\left (4 \, B b c d - 8 \, A c^{2} d - 3 \, B b^{2} e + 4 \, A b c e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*B*x*e/c + (4*B*c*d - 3*B*b*e + 4*A*c*e)/c^2) + 1/8*(4*B*b*c*d - 8*A*c^2*d - 3*B*b^2*e
+ 4*A*b*c*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)

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maple [B]  time = 0.05, size = 202, normalized size = 2.04 \begin {gather*} -\frac {A b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {A d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+\frac {3 B \,b^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {B b d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {\sqrt {c \,x^{2}+b x}\, B e x}{2 c}+\frac {\sqrt {c \,x^{2}+b x}\, A e}{c}-\frac {3 \sqrt {c \,x^{2}+b x}\, B b e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, B d}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*e*x/c*(c*x^2+b*x)^(1/2)-3/4*B*e*b/c^2*(c*x^2+b*x)^(1/2)+3/8*B*e*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^
2+b*x)^(1/2))+1/c*(c*x^2+b*x)^(1/2)*A*e+1/c*(c*x^2+b*x)^(1/2)*B*d-1/2*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+
b*x)^(1/2))*A*e-1/2*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d+A*d*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+
b*x)^(1/2))/c^(1/2)

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maxima [A]  time = 0.70, size = 159, normalized size = 1.61 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B e x}{2 \, c} + \frac {A d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} + \frac {3 \, B b^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} B b e}{4 \, c^{2}} - \frac {{\left (B d + A e\right )} b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} {\left (B d + A e\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2 + b*x)*B*e*x/c + A*d*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/sqrt(c) + 3/8*B*b^2*e*log(2*c
*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/4*sqrt(c*x^2 + b*x)*B*b*e/c^2 - 1/2*(B*d + A*e)*b*log(2*c*x
+ b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(3/2) + sqrt(c*x^2 + b*x)*(B*d + A*e)/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\left (d+e\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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