3.2191 \(\int \frac{x^3}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=107 \[ \frac{2 a^4 x^{3/2}}{3 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{2 a^2 x^{5/2}}{5 b^3}+\frac{2 a^6 \sqrt{x}}{b^7}-\frac{a^5 x}{b^6}-\frac{2 a^7 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{a x^3}{3 b^2}+\frac{2 x^{7/2}}{7 b} \]

[Out]

(2*a^6*Sqrt[x])/b^7 - (a^5*x)/b^6 + (2*a^4*x^(3/2))/(3*b^5) - (a^3*x^2)/(2*b^4) + (2*a^2*x^(5/2))/(5*b^3) - (a
*x^3)/(3*b^2) + (2*x^(7/2))/(7*b) - (2*a^7*Log[a + b*Sqrt[x]])/b^8

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Rubi [A]  time = 0.0697978, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^4 x^{3/2}}{3 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{2 a^2 x^{5/2}}{5 b^3}+\frac{2 a^6 \sqrt{x}}{b^7}-\frac{a^5 x}{b^6}-\frac{2 a^7 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{a x^3}{3 b^2}+\frac{2 x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x]),x]

[Out]

(2*a^6*Sqrt[x])/b^7 - (a^5*x)/b^6 + (2*a^4*x^(3/2))/(3*b^5) - (a^3*x^2)/(2*b^4) + (2*a^2*x^(5/2))/(5*b^3) - (a
*x^3)/(3*b^2) + (2*x^(7/2))/(7*b) - (2*a^7*Log[a + b*Sqrt[x]])/b^8

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^6}{b^7}-\frac{a^5 x}{b^6}+\frac{a^4 x^2}{b^5}-\frac{a^3 x^3}{b^4}+\frac{a^2 x^4}{b^3}-\frac{a x^5}{b^2}+\frac{x^6}{b}-\frac{a^7}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^6 \sqrt{x}}{b^7}-\frac{a^5 x}{b^6}+\frac{2 a^4 x^{3/2}}{3 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{2 a^2 x^{5/2}}{5 b^3}-\frac{a x^3}{3 b^2}+\frac{2 x^{7/2}}{7 b}-\frac{2 a^7 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.059376, size = 99, normalized size = 0.93 \[ \frac{140 a^4 b^3 x^{3/2}-105 a^3 b^4 x^2+84 a^2 b^5 x^{5/2}-210 a^5 b^2 x+420 a^6 b \sqrt{x}-420 a^7 \log \left (a+b \sqrt{x}\right )-70 a b^6 x^3+60 b^7 x^{7/2}}{210 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x]),x]

[Out]

(420*a^6*b*Sqrt[x] - 210*a^5*b^2*x + 140*a^4*b^3*x^(3/2) - 105*a^3*b^4*x^2 + 84*a^2*b^5*x^(5/2) - 70*a*b^6*x^3
+ 60*b^7*x^(7/2) - 420*a^7*Log[a + b*Sqrt[x]])/(210*b^8)

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Maple [A]  time = 0.004, size = 88, normalized size = 0.8 \begin{align*} -{\frac{x{a}^{5}}{{b}^{6}}}+{\frac{2\,{a}^{4}}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}-{\frac{{x}^{2}{a}^{3}}{2\,{b}^{4}}}+{\frac{2\,{a}^{2}}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{2}{7\,b}{x}^{{\frac{7}{2}}}}-2\,{\frac{{a}^{7}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+2\,{\frac{{a}^{6}\sqrt{x}}{{b}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^5*x/b^6+2/3*a^4*x^(3/2)/b^5-1/2*a^3*x^2/b^4+2/5*a^2*x^(5/2)/b^3-1/3*a*x^3/b^2+2/7*x^(7/2)/b-2*a^7*ln(a+b*x^
(1/2))/b^8+2*a^6*x^(1/2)/b^7

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Maxima [A]  time = 0.97533, size = 174, normalized size = 1.63 \begin{align*} -\frac{2 \, a^{7} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{6} a}{3 \, b^{8}} + \frac{42 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{5 \, b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{2 \, b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )}^{3} a^{4}}{3 \, b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{2} a^{5}}{b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )} a^{6}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^7*log(b*sqrt(x) + a)/b^8 + 2/7*(b*sqrt(x) + a)^7/b^8 - 7/3*(b*sqrt(x) + a)^6*a/b^8 + 42/5*(b*sqrt(x) + a)
^5*a^2/b^8 - 35/2*(b*sqrt(x) + a)^4*a^3/b^8 + 70/3*(b*sqrt(x) + a)^3*a^4/b^8 - 21*(b*sqrt(x) + a)^2*a^5/b^8 +
14*(b*sqrt(x) + a)*a^6/b^8

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Fricas [A]  time = 1.26368, size = 215, normalized size = 2.01 \begin{align*} -\frac{70 \, a b^{6} x^{3} + 105 \, a^{3} b^{4} x^{2} + 210 \, a^{5} b^{2} x + 420 \, a^{7} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (15 \, b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 105 \, a^{6} b\right )} \sqrt{x}}{210 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(70*a*b^6*x^3 + 105*a^3*b^4*x^2 + 210*a^5*b^2*x + 420*a^7*log(b*sqrt(x) + a) - 4*(15*b^7*x^3 + 21*a^2*b
^5*x^2 + 35*a^4*b^3*x + 105*a^6*b)*sqrt(x))/b^8

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Sympy [A]  time = 0.983413, size = 109, normalized size = 1.02 \begin{align*} \begin{cases} - \frac{2 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{8}} + \frac{2 a^{6} \sqrt{x}}{b^{7}} - \frac{a^{5} x}{b^{6}} + \frac{2 a^{4} x^{\frac{3}{2}}}{3 b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{2 a^{2} x^{\frac{5}{2}}}{5 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{2 x^{\frac{7}{2}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a**7*log(a/b + sqrt(x))/b**8 + 2*a**6*sqrt(x)/b**7 - a**5*x/b**6 + 2*a**4*x**(3/2)/(3*b**5) - a*
*3*x**2/(2*b**4) + 2*a**2*x**(5/2)/(5*b**3) - a*x**3/(3*b**2) + 2*x**(7/2)/(7*b), Ne(b, 0)), (x**4/(4*a), True
))

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Giac [A]  time = 1.0972, size = 120, normalized size = 1.12 \begin{align*} -\frac{2 \, a^{7} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{60 \, b^{6} x^{\frac{7}{2}} - 70 \, a b^{5} x^{3} + 84 \, a^{2} b^{4} x^{\frac{5}{2}} - 105 \, a^{3} b^{3} x^{2} + 140 \, a^{4} b^{2} x^{\frac{3}{2}} - 210 \, a^{5} b x + 420 \, a^{6} \sqrt{x}}{210 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^7*log(abs(b*sqrt(x) + a))/b^8 + 1/210*(60*b^6*x^(7/2) - 70*a*b^5*x^3 + 84*a^2*b^4*x^(5/2) - 105*a^3*b^3*x
^2 + 140*a^4*b^2*x^(3/2) - 210*a^5*b*x + 420*a^6*sqrt(x))/b^7