3.2.10 \(\int \frac {x^3}{(b \sqrt {x}+a x)^{3/2}} \, dx\) [110]

Optimal. Leaf size=197 \[ -\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{13/2}} \]

[Out]

-693/64*b^5*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/2))/a^(13/2)-4*x^3/a/(b*x^(1/2)+a*x)^(1/2)+693/64*b^4*(
b*x^(1/2)+a*x)^(1/2)/a^6+231/40*b^2*x*(b*x^(1/2)+a*x)^(1/2)/a^4-99/20*b*x^(3/2)*(b*x^(1/2)+a*x)^(1/2)/a^3+22/5
*x^2*(b*x^(1/2)+a*x)^(1/2)/a^2-231/32*b^3*x^(1/2)*(b*x^(1/2)+a*x)^(1/2)/a^5

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Rubi [A]
time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 682, 684, 654, 634, 212} \begin {gather*} -\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{64 a^{13/2}}+\frac {693 b^4 \sqrt {a x+b \sqrt {x}}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{32 a^5}+\frac {231 b^2 x \sqrt {a x+b \sqrt {x}}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {a x+b \sqrt {x}}}{20 a^3}+\frac {22 x^2 \sqrt {a x+b \sqrt {x}}}{5 a^2}-\frac {4 x^3}{a \sqrt {a x+b \sqrt {x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(b*Sqrt[x] + a*x)^(3/2),x]

[Out]

(-4*x^3)/(a*Sqrt[b*Sqrt[x] + a*x]) + (693*b^4*Sqrt[b*Sqrt[x] + a*x])/(64*a^6) - (231*b^3*Sqrt[x]*Sqrt[b*Sqrt[x
] + a*x])/(32*a^5) + (231*b^2*x*Sqrt[b*Sqrt[x] + a*x])/(40*a^4) - (99*b*x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(20*a^3
) + (22*x^2*Sqrt[b*Sqrt[x] + a*x])/(5*a^2) - (693*b^5*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(64*a^
(13/2))

Rule 212

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

Rule 634

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 654

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 682

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

Rule 684

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 2043

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=2 \text {Subst}\left (\int \frac {x^7}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {22 \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {(99 b) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{5 a^2}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}+\frac {\left (693 b^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{40 a^3}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (231 b^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{16 a^4}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}+\frac {\left (693 b^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^5}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (693 b^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{128 a^6}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (693 b^5\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^6}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 137, normalized size = 0.70 \begin {gather*} \frac {\sqrt {b \sqrt {x}+a x} \left (3465 b^5+1155 a b^4 \sqrt {x}-462 a^2 b^3 x+264 a^3 b^2 x^{3/2}-176 a^4 b x^2+128 a^5 x^{5/2}\right )}{320 a^6 \left (b+a \sqrt {x}\right )}+\frac {693 b^5 \log \left (b+2 a \sqrt {x}-2 \sqrt {a} \sqrt {b \sqrt {x}+a x}\right )}{128 a^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b*Sqrt[x] + a*x)^(3/2),x]

[Out]

(Sqrt[b*Sqrt[x] + a*x]*(3465*b^5 + 1155*a*b^4*Sqrt[x] - 462*a^2*b^3*x + 264*a^3*b^2*x^(3/2) - 176*a^4*b*x^2 +
128*a^5*x^(5/2)))/(320*a^6*(b + a*Sqrt[x])) + (693*b^5*Log[b + 2*a*Sqrt[x] - 2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]])
/(128*a^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(147)=294\).
time = 0.39, size = 549, normalized size = 2.79

method	result	size
derivativedivides	\(\frac {2 x^{3}}{5 a \sqrt {b \sqrt {x}+a x}}-\frac {11 b \left (\frac {x^{\frac {5}{2}}}{4 a \sqrt {b \sqrt {x}+a x}}-\frac {9 b \left (\frac {x^{2}}{3 a \sqrt {b \sqrt {x}+a x}}-\frac {7 b \left (\frac {x^{\frac {3}{2}}}{2 a \sqrt {b \sqrt {x}+a x}}-\frac {5 b \left (\frac {x}{a \sqrt {b \sqrt {x}+a x}}-\frac {3 b \left (-\frac {\sqrt {x}}{a \sqrt {b \sqrt {x}+a x}}-\frac {b \left (-\frac {1}{a \sqrt {b \sqrt {x}+a x}}+\frac {b +2 a \sqrt {x}}{b a \sqrt {b \sqrt {x}+a x}}\right )}{2 a}+\frac {\ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{5 a}\)	\(227\)
default	\(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (352 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {3}{2}} a^{\frac {11}{2}} b -256 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{2} a^{\frac {13}{2}}-528 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x \,a^{\frac {9}{2}} b^{2}+4060 \sqrt {b \sqrt {x}+a x}\, x^{\frac {3}{2}} a^{\frac {9}{2}} b^{3}-3136 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {x}\, a^{\frac {7}{2}} b^{3}+10150 \sqrt {b \sqrt {x}+a x}\, x \,a^{\frac {7}{2}} b^{4}-8960 x \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {7}{2}} b^{4}+4480 x \,a^{3} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{5}-2000 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4}+8120 \sqrt {b \sqrt {x}+a x}\, \sqrt {x}\, a^{\frac {5}{2}} b^{5}-1015 x \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{5}-17920 \sqrt {x}\, \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {5}{2}} b^{5}+8960 \sqrt {x}\, a^{2} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{6}+2560 \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4}+2030 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b^{6}-2030 \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b^{6}-8960 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {3}{2}} b^{6}+4480 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{7}-1015 \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{7}\right )}{640 a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \left (a \sqrt {x}+b \right )^{2}}\)	\(549\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^(1/2)+a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/640*(b*x^(1/2)+a*x)^(1/2)/a^(15/2)*(352*(b*x^(1/2)+a*x)^(3/2)*x^(3/2)*a^(11/2)*b-256*(b*x^(1/2)+a*x)^(3/2)*
x^2*a^(13/2)-528*(b*x^(1/2)+a*x)^(3/2)*x*a^(9/2)*b^2+4060*(b*x^(1/2)+a*x)^(1/2)*x^(3/2)*a^(9/2)*b^3-3136*(b*x^
(1/2)+a*x)^(3/2)*x^(1/2)*a^(7/2)*b^3+10150*(b*x^(1/2)+a*x)^(1/2)*x*a^(7/2)*b^4-8960*x*(x^(1/2)*(a*x^(1/2)+b))^
(1/2)*a^(7/2)*b^4+4480*x*a^3*ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*b^5-2000*
(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*b^4+8120*(b*x^(1/2)+a*x)^(1/2)*x^(1/2)*a^(5/2)*b^5-1015*x*ln(1/2*(2*a*x^(1/2)+2*
(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a^3*b^5-17920*x^(1/2)*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(5/2)*b^5+8960
*x^(1/2)*a^2*ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*b^6+2560*(x^(1/2)*(a*x^(1
/2)+b))^(3/2)*a^(5/2)*b^4+2030*(b*x^(1/2)+a*x)^(1/2)*a^(3/2)*b^6-2030*x^(1/2)*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)
+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a^2*b^6-8960*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(3/2)*b^6+4480*a*ln(1/2*(2*a*x^(1
/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*b^7-1015*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a
^(1/2)+b)/a^(1/2))*a*b^7)/(x^(1/2)*(a*x^(1/2)+b))^(1/2)/(a*x^(1/2)+b)^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(x^3/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3/(a*x + b*sqrt(x))^(3/2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

Integral(x**3/(a*x + b*sqrt(x))**(3/2), x)

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Giac [A]
time = 0.80, size = 150, normalized size = 0.76 \begin {gather*} \frac {1}{320} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, \sqrt {x} {\left (\frac {8 \, \sqrt {x}}{a^{2}} - \frac {19 \, b}{a^{3}}\right )} + \frac {71 \, b^{2}}{a^{4}}\right )} \sqrt {x} - \frac {515 \, b^{3}}{a^{5}}\right )} \sqrt {x} + \frac {2185 \, b^{4}}{a^{6}}\right )} + \frac {693 \, b^{5} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{128 \, a^{\frac {13}{2}}} + \frac {4 \, b^{6}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*sqrt(a*x + b*sqrt(x))*(2*(4*(2*sqrt(x)*(8*sqrt(x)/a^2 - 19*b/a^3) + 71*b^2/a^4)*sqrt(x) - 515*b^3/a^5)*s
qrt(x) + 2185*b^4/a^6) + 693/128*b^5*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^(13/
2) + 4*b^6/((a*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + sqrt(a)*b)*a^6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x + b*x^(1/2))^(3/2),x)

[Out]

int(x^3/(a*x + b*x^(1/2))^(3/2), x)

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