3.109 \(\int \frac{x^4 (A+B x)}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=197 \[ \frac{7 b^3 \sqrt{b x+c x^2} (9 b B-10 A c)}{128 c^5}-\frac{7 b^2 x \sqrt{b x+c x^2} (9 b B-10 A c)}{192 c^4}-\frac{7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{11/2}}+\frac{7 b x^2 \sqrt{b x+c x^2} (9 b B-10 A c)}{240 c^3}-\frac{x^3 \sqrt{b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c} \]

[Out]

(7*b^3*(9*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(128*c^5) - (7*b^2*(9*b*B - 10*A*c)*x*Sqrt[b*x + c*x^2])/(192*c^4)
+ (7*b*(9*b*B - 10*A*c)*x^2*Sqrt[b*x + c*x^2])/(240*c^3) - ((9*b*B - 10*A*c)*x^3*Sqrt[b*x + c*x^2])/(40*c^2) +
(B*x^4*Sqrt[b*x + c*x^2])/(5*c) - (7*b^4*(9*b*B - 10*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(11/
2))

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Rubi [A]  time = 0.200275, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {794, 670, 640, 620, 206} \[ \frac{7 b^3 \sqrt{b x+c x^2} (9 b B-10 A c)}{128 c^5}-\frac{7 b^2 x \sqrt{b x+c x^2} (9 b B-10 A c)}{192 c^4}-\frac{7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{11/2}}+\frac{7 b x^2 \sqrt{b x+c x^2} (9 b B-10 A c)}{240 c^3}-\frac{x^3 \sqrt{b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/Sqrt[b*x + c*x^2],x]

[Out]

(7*b^3*(9*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(128*c^5) - (7*b^2*(9*b*B - 10*A*c)*x*Sqrt[b*x + c*x^2])/(192*c^4)
+ (7*b*(9*b*B - 10*A*c)*x^2*Sqrt[b*x + c*x^2])/(240*c^3) - ((9*b*B - 10*A*c)*x^3*Sqrt[b*x + c*x^2])/(40*c^2) +
(B*x^4*Sqrt[b*x + c*x^2])/(5*c) - (7*b^4*(9*b*B - 10*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(11/
2))

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), 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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 670

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[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

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

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 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])

Rubi steps

\begin{align*} \int \frac{x^4 (A+B x)}{\sqrt{b x+c x^2}} \, dx &=\frac{B x^4 \sqrt{b x+c x^2}}{5 c}+\frac{\left (4 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \int \frac{x^4}{\sqrt{b x+c x^2}} \, dx}{5 c}\\ &=-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}+\frac{(7 b (9 b B-10 A c)) \int \frac{x^3}{\sqrt{b x+c x^2}} \, dx}{80 c^2}\\ &=\frac{7 b (9 b B-10 A c) x^2 \sqrt{b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}-\frac{\left (7 b^2 (9 b B-10 A c)\right ) \int \frac{x^2}{\sqrt{b x+c x^2}} \, dx}{96 c^3}\\ &=-\frac{7 b^2 (9 b B-10 A c) x \sqrt{b x+c x^2}}{192 c^4}+\frac{7 b (9 b B-10 A c) x^2 \sqrt{b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}+\frac{\left (7 b^3 (9 b B-10 A c)\right ) \int \frac{x}{\sqrt{b x+c x^2}} \, dx}{128 c^4}\\ &=\frac{7 b^3 (9 b B-10 A c) \sqrt{b x+c x^2}}{128 c^5}-\frac{7 b^2 (9 b B-10 A c) x \sqrt{b x+c x^2}}{192 c^4}+\frac{7 b (9 b B-10 A c) x^2 \sqrt{b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}-\frac{\left (7 b^4 (9 b B-10 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^5}\\ &=\frac{7 b^3 (9 b B-10 A c) \sqrt{b x+c x^2}}{128 c^5}-\frac{7 b^2 (9 b B-10 A c) x \sqrt{b x+c x^2}}{192 c^4}+\frac{7 b (9 b B-10 A c) x^2 \sqrt{b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}-\frac{\left (7 b^4 (9 b B-10 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{128 c^5}\\ &=\frac{7 b^3 (9 b B-10 A c) \sqrt{b x+c x^2}}{128 c^5}-\frac{7 b^2 (9 b B-10 A c) x \sqrt{b x+c x^2}}{192 c^4}+\frac{7 b (9 b B-10 A c) x^2 \sqrt{b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{b x+c x^2}}{5 c}-\frac{7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.364264, size = 133, normalized size = 0.68 \[ \frac{\sqrt{x (b+c x)} \left (\frac{(9 b B-10 A c) \left (c x \sqrt{\frac{c x}{b}+1} \left (-70 b^2 c x+105 b^3+56 b c^2 x^2-48 c^3 x^3\right )-105 b^{7/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{\sqrt{\frac{c x}{b}+1}}+384 B c^5 x^5\right )}{1920 c^6 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[x*(b + c*x)]*(384*B*c^5*x^5 + ((9*b*B - 10*A*c)*(c*x*Sqrt[1 + (c*x)/b]*(105*b^3 - 70*b^2*c*x + 56*b*c^2*
x^2 - 48*c^3*x^3) - 105*b^(7/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/Sqrt[1 + (c*x)/b]))/(1920
*c^6*x)

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Maple [A]  time = 0.009, size = 255, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}B}{5\,c}\sqrt{c{x}^{2}+bx}}-{\frac{9\,bB{x}^{3}}{40\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{21\,{b}^{2}B{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{21\,{b}^{3}Bx}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{63\,{b}^{4}B}{128\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{63\,B{b}^{5}}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}}+{\frac{A{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{7\,Ab{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,A{b}^{2}x}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{35\,A{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,A{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(c*x^2+b*x)^(1/2),x)

[Out]

1/5*B*x^4*(c*x^2+b*x)^(1/2)/c-9/40*B*b/c^2*x^3*(c*x^2+b*x)^(1/2)+21/80*B*b^2/c^3*x^2*(c*x^2+b*x)^(1/2)-21/64*B
*b^3/c^4*x*(c*x^2+b*x)^(1/2)+63/128*B*b^4/c^5*(c*x^2+b*x)^(1/2)-63/256*B*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x)^(1/2))+1/4*A*x^3/c*(c*x^2+b*x)^(1/2)-7/24*A*b/c^2*x^2*(c*x^2+b*x)^(1/2)+35/96*A*b^2/c^3*x*(c*x^2+b*
x)^(1/2)-35/64*A*b^3/c^4*(c*x^2+b*x)^(1/2)+35/128*A*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.95697, size = 721, normalized size = 3.66 \begin{align*} \left [-\frac{105 \,{\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c - 1050 \, A b^{3} c^{2} - 48 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} + 56 \,{\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} - 70 \,{\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{3840 \, c^{6}}, \frac{105 \,{\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c - 1050 \, A b^{3} c^{2} - 48 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} + 56 \,{\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} - 70 \,{\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{1920 \, c^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

[-1/3840*(105*(9*B*b^5 - 10*A*b^4*c)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(384*B*c^5*x^4 +
945*B*b^4*c - 1050*A*b^3*c^2 - 48*(9*B*b*c^4 - 10*A*c^5)*x^3 + 56*(9*B*b^2*c^3 - 10*A*b*c^4)*x^2 - 70*(9*B*b^
3*c^2 - 10*A*b^2*c^3)*x)*sqrt(c*x^2 + b*x))/c^6, 1/1920*(105*(9*B*b^5 - 10*A*b^4*c)*sqrt(-c)*arctan(sqrt(c*x^2
+ b*x)*sqrt(-c)/(c*x)) + (384*B*c^5*x^4 + 945*B*b^4*c - 1050*A*b^3*c^2 - 48*(9*B*b*c^4 - 10*A*c^5)*x^3 + 56*(
9*B*b^2*c^3 - 10*A*b*c^4)*x^2 - 70*(9*B*b^3*c^2 - 10*A*b^2*c^3)*x)*sqrt(c*x^2 + b*x))/c^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (A + B x\right )}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**4*(A + B*x)/sqrt(x*(b + c*x)), x)

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Giac [A]  time = 1.24597, size = 223, normalized size = 1.13 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, B x}{c} - \frac{9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac{7 \,{\left (9 \, B b^{2} c^{2} - 10 \, A b c^{3}\right )}}{c^{5}}\right )} x - \frac{35 \,{\left (9 \, B b^{3} c - 10 \, A b^{2} c^{2}\right )}}{c^{5}}\right )} x + \frac{105 \,{\left (9 \, B b^{4} - 10 \, A b^{3} c\right )}}{c^{5}}\right )} + \frac{7 \,{\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x)*(2*(4*(6*(8*B*x/c - (9*B*b*c^3 - 10*A*c^4)/c^5)*x + 7*(9*B*b^2*c^2 - 10*A*b*c^3)/c^5)
*x - 35*(9*B*b^3*c - 10*A*b^2*c^2)/c^5)*x + 105*(9*B*b^4 - 10*A*b^3*c)/c^5) + 7/256*(9*B*b^5 - 10*A*b^4*c)*log
(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)