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

Optimal. Leaf size=51 \[ \frac{6}{5} a^2 b x^{5/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{9} b^3 x^{9/2} \]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(9/2))/9

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Rubi [A]  time = 0.0111421, antiderivative size = 51, 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} \[ \frac{6}{5} a^2 b x^{5/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^3,x]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(9/2))/9

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 \sqrt{x} (a+b x)^3 \, dx &=\int \left (a^3 \sqrt{x}+3 a^2 b x^{3/2}+3 a b^2 x^{5/2}+b^3 x^{7/2}\right ) \, dx\\ &=\frac{2}{3} a^3 x^{3/2}+\frac{6}{5} a^2 b x^{5/2}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{9} b^3 x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0105123, size = 39, normalized size = 0.76 \[ \frac{2}{315} x^{3/2} \left (189 a^2 b x+105 a^3+135 a b^2 x^2+35 b^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x)^3,x]

[Out]

(2*x^(3/2)*(105*a^3 + 189*a^2*b*x + 135*a*b^2*x^2 + 35*b^3*x^3))/315

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Maple [A]  time = 0.003, size = 36, normalized size = 0.7 \begin{align*}{\frac{70\,{b}^{3}{x}^{3}+270\,a{b}^{2}{x}^{2}+378\,{a}^{2}bx+210\,{a}^{3}}{315}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*x^(3/2)*(35*b^3*x^3+135*a*b^2*x^2+189*a^2*b*x+105*a^3)

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Maxima [A]  time = 1.06095, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)

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Fricas [A]  time = 1.49454, size = 97, normalized size = 1.9 \begin{align*} \frac{2}{315} \,{\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^3*x^4 + 135*a*b^2*x^3 + 189*a^2*b*x^2 + 105*a^3*x)*sqrt(x)

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Sympy [C]  time = 4.84307, size = 4886, normalized size = 95.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-32*a**(49/2)*sqrt(-1 + b*(a/b + x)/a)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a
**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15
*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 32*I*a**(49/2)/(315*a**20*b**(3/2) - 1890*a**19*
b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2
)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 176*a**(47/2)*b*sqrt(
-1 + b*(a/b + x)/a)*(a/b + x)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b +
 x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)
**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 192*I*a**(47/2)*b*(a/b + x)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2
)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b
+ x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 396*a**(45/2)*b**2*sqrt(-1 +
 b*(a/b + x)/a)*(a/b + x)**2/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b +
x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)*
*5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 480*I*a**(45/2)*b**2*(a/b + x)**2/(315*a**20*b**(3/2) - 1890*a**19*b*
*(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*
(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 462*a**(43/2)*b**3*sqrt
(-1 + b*(a/b + x)/a)*(a/b + x)**3/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a
/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b
+ x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 640*I*a**(43/2)*b**3*(a/b + x)**3/(315*a**20*b**(3/2) - 1890*a**
19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(1
1/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 210*a**(41/2)*b**4
*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**4/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/
2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*
(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 480*I*a**(41/2)*b**4*(a/b + x)**4/(315*a**20*b**(3/2) - 189
0*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*
b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 378*a**(39/2)
*b**5*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**5/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b
**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(1
3/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 192*I*a**(39/2)*b**5*(a/b + x)**5/(315*a**20*b**(3/2)
- 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a
**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 1134*a**
(37/2)*b**6*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**6/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a
**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15
*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 32*I*a**(37/2)*b**6*(a/b + x)**6/(315*a**20*b**(
3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4
725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 149
4*a**(35/2)*b**7*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**7/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4
725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*
a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 1098*a**(33/2)*b**8*sqrt(-1 + b*(a/b + x)/a
)*(a/b + x)**8/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a
**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14
*b**(15/2)*(a/b + x)**6) - 430*a**(31/2)*b**9*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**9/(315*a**20*b**(3/2) - 1890
*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b
**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 70*a**(29/2)*b
**10*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**10/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b
**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(1
3/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6), Abs(b*(a/b + x))/Abs(a) > 1), (-32*I*a**(49/2)*sqrt(1 -
 b*(a/b + x)/a)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*
a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**1
4*b**(15/2)*(a/b + x)**6) + 32*I*a**(49/2)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b*
*(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13
/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 176*I*a**(47/2)*b*sqrt(1 - b*(a/b + x)/a)*(a/b + x)/(31
5*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b
 + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b +
x)**6) - 192*I*a**(47/2)*b*(a/b + x)/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)
*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a
/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 396*I*a**(45/2)*b**2*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**2/(31
5*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b
 + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b +
x)**6) + 480*I*a**(45/2)*b**2*(a/b + x)**2/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b*
*(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13
/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 462*I*a**(43/2)*b**3*sqrt(1 - b*(a/b + x)/a)*(a/b + x)*
*3/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2
)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(
a/b + x)**6) - 640*I*a**(43/2)*b**3*(a/b + x)**3/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a*
*18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*
b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 210*I*a**(41/2)*b**4*sqrt(1 - b*(a/b + x)/a)*(a/b
 + x)**4/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b
**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(1
5/2)*(a/b + x)**6) + 480*I*a**(41/2)*b**4*(a/b + x)**4/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4
725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*
a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 378*I*a**(39/2)*b**5*sqrt(1 - b*(a/b + x)/a
)*(a/b + x)**5/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a
**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14
*b**(15/2)*(a/b + x)**6) - 192*I*a**(39/2)*b**5*(a/b + x)**5/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b +
x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 -
 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 1134*I*a**(37/2)*b**6*sqrt(1 - b*(a/b
 + x)/a)*(a/b + x)**6/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 -
 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 31
5*a**14*b**(15/2)*(a/b + x)**6) + 32*I*a**(37/2)*b**6*(a/b + x)**6/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(
a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x
)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) - 1494*I*a**(35/2)*b**7*sqrt(1 -
b*(a/b + x)/a)*(a/b + x)**7/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x
)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**
5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 1098*I*a**(33/2)*b**8*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**8/(315*a**20*
b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3
 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) -
 430*I*a**(31/2)*b**9*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**9/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x)
 + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1
890*a**15*b**(13/2)*(a/b + x)**5 + 315*a**14*b**(15/2)*(a/b + x)**6) + 70*I*a**(29/2)*b**10*sqrt(1 - b*(a/b +
x)/a)*(a/b + x)**10/(315*a**20*b**(3/2) - 1890*a**19*b**(5/2)*(a/b + x) + 4725*a**18*b**(7/2)*(a/b + x)**2 - 6
300*a**17*b**(9/2)*(a/b + x)**3 + 4725*a**16*b**(11/2)*(a/b + x)**4 - 1890*a**15*b**(13/2)*(a/b + x)**5 + 315*
a**14*b**(15/2)*(a/b + x)**6), True))

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Giac [A]  time = 1.20288, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)