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

Optimal. Leaf size=68 \[ \frac{2 a^2 (a+b x)^{3/2}}{b^4}-\frac{2 a^3 \sqrt{a+b x}}{b^4}+\frac{2 (a+b x)^{7/2}}{7 b^4}-\frac{6 a (a+b x)^{5/2}}{5 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0178664, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2 a^2 (a+b x)^{3/2}}{b^4}-\frac{2 a^3 \sqrt{a+b x}}{b^4}+\frac{2 (a+b x)^{7/2}}{7 b^4}-\frac{6 a (a+b x)^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0385522, size = 46, normalized size = 0.68 \[ \frac{2 \sqrt{a+b x} \left (8 a^2 b x-16 a^3-6 a b^2 x^2+5 b^3 x^3\right )}{35 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(-16*a^3 + 8*a^2*b*x - 6*a*b^2*x^2 + 5*b^3*x^3))/(35*b^4)

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Maple [A]  time = 0.004, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-10\,{b}^{3}{x}^{3}+12\,a{b}^{2}{x}^{2}-16\,{a}^{2}bx+32\,{a}^{3}}{35\,{b}^{4}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(b*x+a)^(1/2)*(-5*b^3*x^3+6*a*b^2*x^2-8*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.03788, size = 76, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{4}} - \frac{6 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{2 \, \sqrt{b x + a} a^{3}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52225, size = 96, normalized size = 1.41 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 8 \, a^{2} b x - 16 \, a^{3}\right )} \sqrt{b x + a}}{35 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b^3*x^3 - 6*a*b^2*x^2 + 8*a^2*b*x - 16*a^3)*sqrt(b*x + a)/b^4

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Sympy [B]  time = 3.22659, size = 1640, normalized size = 24.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32*a**(47/2)*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 +
525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 32*a**(47/2)/(35*a**20*b**4 + 210*a**19*b**
5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x
**6) - 176*a**(45/2)*b*x*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b
**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 192*a**(45/2)*b*x/(35*a**20*b**4
+ 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 +
35*a**14*b**10*x**6) - 396*a**(43/2)*b**2*x**2*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b
**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 480*a**(43
/2)*b**2*x**2/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x
**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) - 462*a**(41/2)*b**3*x**3*sqrt(1 + b*x/a)/(35*a**20*b**4 + 21
0*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a*
*14*b**10*x**6) + 640*a**(41/2)*b**3*x**3/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*
b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) - 280*a**(39/2)*b**4*x**4*sqrt(1
+ b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 +
210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 480*a**(39/2)*b**4*x**4/(35*a**20*b**4 + 210*a**19*b**5*x + 525*
a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) - 42*
a**(37/2)*b**5*x**5*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x
**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 192*a**(37/2)*b**5*x**5/(35*a**20*b**
4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 +
35*a**14*b**10*x**6) + 84*a**(35/2)*b**6*x**6*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b
**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 32*a**(35/
2)*b**6*x**6/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x*
*4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 94*a**(33/2)*b**7*x**7*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*
a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**1
4*b**10*x**6) + 48*a**(31/2)*b**8*x**8*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2
+ 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 10*a**(29/2)*b**9*
x**9*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16
*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6)

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Giac [A]  time = 1.19916, size = 66, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )}}{35 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)/b^4