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

Optimal. Leaf size=47 \[ 2 a^3 \sqrt{x}+2 a^2 b x^{3/2}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0297594, antiderivative size = 47, 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 \[ 2 a^3 \sqrt{x}+2 a^2 b x^{3/2}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.1725, size = 46, normalized size = 0.98 \[ 2 a^{3} \sqrt{x} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0101665, size = 39, normalized size = 0.83 \[ \frac{2}{35} \sqrt{x} \left (35 a^3+35 a^2 b x+21 a b^2 x^2+5 b^3 x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(35*a^3 + 35*a^2*b*x + 21*a*b^2*x^2 + 5*b^3*x^3))/35

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[{\frac{10\,{b}^{3}{x}^{3}+42\,a{b}^{2}{x}^{2}+70\,{a}^{2}bx+70\,{a}^{3}}{35}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^(1/2),x)

[Out]

2/35*x^(1/2)*(5*b^3*x^3+21*a*b^2*x^2+35*a^2*b*x+35*a^3)

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Maxima [A]  time = 1.31838, size = 47, normalized size = 1. \[ \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/sqrt(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205526, size = 47, normalized size = 1. \[ \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 35 \, a^{2} b x + 35 \, a^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/sqrt(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.37274, size = 4600, normalized size = 97.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((32*a**(47/2)*sqrt(-1 + b*(a/b + x)/a)/(35*a**20*sqrt(b) - 210*a**19*b
**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b +
x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*
a**14*b**(13/2)*(a/b + x)**6) - 32*I*a**(47/2)/(35*a**20*sqrt(b) - 210*a**19*b**
(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)
**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a*
*14*b**(13/2)*(a/b + x)**6) - 176*a**(45/2)*b*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)
/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)
**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a*
*15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 192*I*a**(45/2)*
b*(a/b + x)/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2
)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)*
*4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 396*a
**(43/2)*b**2*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**2/(35*a**20*sqrt(b) - 210*a**1
9*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b
+ x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 +
35*a**14*b**(13/2)*(a/b + x)**6) - 480*I*a**(43/2)*b**2*(a/b + x)**2/(35*a**20*s
qrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a*
*17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2
)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 462*a**(41/2)*b**3*sqrt(-1 +
b*(a/b + x)/a)*(a/b + x)**3/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) +
525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b*
*(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b
+ x)**6) + 640*I*a**(41/2)*b**3*(a/b + x)**3/(35*a**20*sqrt(b) - 210*a**19*b**(
3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)*
*3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**
14*b**(13/2)*(a/b + x)**6) + 280*a**(39/2)*b**4*sqrt(-1 + b*(a/b + x)/a)*(a/b +
x)**4/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b
+ x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 2
10*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 480*I*a**(3
9/2)*b**4*(a/b + x)**4/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a*
*18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)
*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)*
*6) - 42*a**(37/2)*b**5*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**5/(35*a**20*sqrt(b)
- 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**
(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b
+ x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 192*I*a**(37/2)*b**5*(a/b + x)**5/(
35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**
2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**1
5*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 84*a**(35/2)*b**6*
sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**6/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/
b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525
*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(1
3/2)*(a/b + x)**6) - 32*I*a**(35/2)*b**6*(a/b + x)**6/(35*a**20*sqrt(b) - 210*a*
*19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a
/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5
+ 35*a**14*b**(13/2)*(a/b + x)**6) + 94*a**(33/2)*b**7*sqrt(-1 + b*(a/b + x)/a)*
(a/b + x)**7/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/
2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)
**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 48*a
**(31/2)*b**8*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**8/(35*a**20*sqrt(b) - 210*a**1
9*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b
+ x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 +
35*a**14*b**(13/2)*(a/b + x)**6) + 10*a**(29/2)*b**9*sqrt(-1 + b*(a/b + x)/a)*(a
/b + x)**9/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)
*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**
4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6), Abs(b*(
a/b + x)/a) > 1), (32*I*a**(47/2)*sqrt(1 - b*(a/b + x)/a)/(35*a**20*sqrt(b) - 21
0*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2
)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)
**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 32*I*a**(47/2)/(35*a**20*sqrt(b) - 210*
a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*
(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**
5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 176*I*a**(45/2)*b*sqrt(1 - b*(a/b + x)/a)
*(a/b + x)/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)
*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**
4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 192*I*
a**(45/2)*b*(a/b + x)/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**
18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*
(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**
6) + 396*I*a**(43/2)*b**2*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**2/(35*a**20*sqrt(b)
- 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b*
*(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b
+ x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 480*I*a**(43/2)*b**2*(a/b + x)**2/
(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)*
*2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**
15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 462*I*a**(41/2)*b
**3*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**3/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*
(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 +
525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b*
*(13/2)*(a/b + x)**6) + 640*I*a**(41/2)*b**3*(a/b + x)**3/(35*a**20*sqrt(b) - 21
0*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2
)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)
**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 280*I*a**(39/2)*b**4*sqrt(1 - b*(a/b +
x)/a)*(a/b + x)**4/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*
b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/
b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6)
- 480*I*a**(39/2)*b**4*(a/b + x)**4/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b
+ x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a
**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/
2)*(a/b + x)**6) - 42*I*a**(37/2)*b**5*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**5/(35*
a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 -
700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b
**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 192*I*a**(37/2)*b**5*
(a/b + x)**5/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/
2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)
**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) - 84*I
*a**(35/2)*b**6*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**6/(35*a**20*sqrt(b) - 210*a**
19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/
b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 +
35*a**14*b**(13/2)*(a/b + x)**6) - 32*I*a**(35/2)*b**6*(a/b + x)**6/(35*a**20*s
qrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a*
*17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2
)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 94*I*a**(33/2)*b**7*sqrt(1 -
b*(a/b + x)/a)*(a/b + x)**7/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) +
525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b*
*(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b
+ x)**6) - 48*I*a**(31/2)*b**8*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**8/(35*a**20*s
qrt(b) - 210*a**19*b**(3/2)*(a/b + x) + 525*a**18*b**(5/2)*(a/b + x)**2 - 700*a*
*17*b**(7/2)*(a/b + x)**3 + 525*a**16*b**(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2
)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b + x)**6) + 10*I*a**(29/2)*b**9*sqrt(1 -
b*(a/b + x)/a)*(a/b + x)**9/(35*a**20*sqrt(b) - 210*a**19*b**(3/2)*(a/b + x) +
525*a**18*b**(5/2)*(a/b + x)**2 - 700*a**17*b**(7/2)*(a/b + x)**3 + 525*a**16*b*
*(9/2)*(a/b + x)**4 - 210*a**15*b**(11/2)*(a/b + x)**5 + 35*a**14*b**(13/2)*(a/b
+ x)**6), True))

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GIAC/XCAS [A]  time = 0.202563, size = 47, normalized size = 1. \[ \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/sqrt(x),x, algorithm="giac")

[Out]

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