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

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

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Rubi [A]  time = 0.20, antiderivative size = 197, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.35, size = 133, normalized size = 0.68 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {(9 b B-10 A c) \left (c x \sqrt {\frac {c x}{b}+1} \left (105 b^3-70 b^2 c x+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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.62, size = 153, normalized size = 0.78 \begin {gather*} \frac {7 \left (9 b^5 B-10 A b^4 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{256 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (-1050 A b^3 c+700 A b^2 c^2 x-560 A b c^3 x^2+480 A c^4 x^3+945 b^4 B-630 b^3 B c x+504 b^2 B c^2 x^2-432 b B c^3 x^3+384 B c^4 x^4\right )}{1920 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(945*b^4*B - 1050*A*b^3*c - 630*b^3*B*c*x + 700*A*b^2*c^2*x + 504*b^2*B*c^2*x^2 - 560*A*b*c
^3*x^2 - 432*b*B*c^3*x^3 + 480*A*c^4*x^3 + 384*B*c^4*x^4))/(1920*c^5) + (7*(9*b^5*B - 10*A*b^4*c)*Log[b + 2*c*
x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(256*c^(11/2))

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fricas [A]  time = 0.44, size = 303, normalized size = 1.54 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.23, size = 165, normalized size = 0.84 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 255, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, B \,x^{4}}{5 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,x^{3}}{4 c}-\frac {9 \sqrt {c \,x^{2}+b x}\, B b \,x^{3}}{40 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A b \,x^{2}}{24 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x}\, B \,b^{2} x^{2}}{80 c^{3}}+\frac {35 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}-\frac {63 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {11}{2}}}+\frac {35 \sqrt {c \,x^{2}+b x}\, A \,b^{2} x}{96 c^{3}}-\frac {21 \sqrt {c \,x^{2}+b x}\, B \,b^{3} x}{64 c^{4}}-\frac {35 \sqrt {c \,x^{2}+b x}\, A \,b^{3}}{64 c^{4}}+\frac {63 \sqrt {c \,x^{2}+b x}\, B \,b^{4}}{128 c^{5}} \end {gather*}

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((c*x+1/2*b)/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((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))

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maxima [A]  time = 0.96, size = 252, normalized size = 1.28 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B x^{4}}{5 \, c} - \frac {9 \, \sqrt {c x^{2} + b x} B b x^{3}}{40 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A x^{3}}{4 \, c} + \frac {21 \, \sqrt {c x^{2} + b x} B b^{2} x^{2}}{80 \, c^{3}} - \frac {7 \, \sqrt {c x^{2} + b x} A b x^{2}}{24 \, c^{2}} - \frac {21 \, \sqrt {c x^{2} + b x} B b^{3} x}{64 \, c^{4}} + \frac {35 \, \sqrt {c x^{2} + b x} A b^{2} x}{96 \, c^{3}} - \frac {63 \, B b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {11}{2}}} + \frac {35 \, A b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} + \frac {63 \, \sqrt {c x^{2} + b x} B b^{4}}{128 \, c^{5}} - \frac {35 \, \sqrt {c x^{2} + b x} A b^{3}}{64 \, c^{4}} \end {gather*}

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]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^4*(A + B*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 {x^{4} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}

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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