3.4.35 \(\int \frac {x^3}{\sqrt {a+b x}} \, dx\) [335]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} -\frac {2 a^3 \sqrt {a+b x}}{b^4}+\frac {2 a^2 (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{7/2}}{7 b^4}-\frac {6 a (a+b x)^{5/2}}{5 b^4} \end {gather*}

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 45

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*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.68 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-16 a^3+8 a^2 b x-6 a b^2 x^2+5 b^3 x^3\right )}{35 b^4} \end {gather*}

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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(330\) vs. \(2(68)=136\).
time = 15.24, size = 308, normalized size = 4.53 \begin {gather*} \frac {2 \sqrt {a} \left (16 a^9 \left (1-\sqrt {\frac {a+b x}{a}}\right )+8 a^8 b x \left (12-11 \sqrt {\frac {a+b x}{a}}\right )+6 a^7 b^2 x^2 \left (40-33 \sqrt {\frac {a+b x}{a}}\right )+a^6 b^3 x^3 \left (320-231 \sqrt {\frac {a+b x}{a}}\right )+5 b^4 x^4 \left (48 a^5+b^5 x^5 \sqrt {\frac {a+b x}{a}}\right )-140 a^5 b^4 x^4 \sqrt {\frac {a+b x}{a}}+a b^5 x^5 \left (96 a^3+16 a^2 b x+47 a b^2 x^2 \sqrt {\frac {a+b x}{a}}+24 b^3 x^3 \sqrt {\frac {a+b x}{a}}\right )+21 a^3 b^5 x^5 \left (-a+2 b x\right ) \sqrt {\frac {a+b x}{a}}\right )}{35 b^4 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2 Sqrt[a] (16 a ^ 9 (1 - Sqrt[(a + b x) / a]) + 8 a ^ 8 b x (12 - 11 Sqrt[(a + b x) / a]) + 6 a ^ 7 b ^ 2 x ^
2 (40 - 33 Sqrt[(a + b x) / a]) + a ^ 6 b ^ 3 x ^ 3 (320 - 231 Sqrt[(a + b x) / a]) + 5 b ^ 4 x ^ 4 (48 a ^ 5
+ b ^ 5 x ^ 5 Sqrt[(a + b x) / a]) - 140 a ^ 5 b ^ 4 x ^ 4 Sqrt[(a + b x) / a] + a b ^ 5 x ^ 5 (96 a ^ 3 + 16
a ^ 2 b x + 47 a b ^ 2 x ^ 2 Sqrt[(a + b x) / a] + 24 b ^ 3 x ^ 3 Sqrt[(a + b x) / a]) + 21 a ^ 3 b ^ 5 x ^ 5
(-a + 2 b x) Sqrt[(a + b x) / a]) / (35 b ^ 4 (a ^ 6 + 6 a ^ 5 b x + 15 a ^ 4 b ^ 2 x ^ 2 + 20 a ^ 3 b ^ 3 x ^
3 + 15 a ^ 2 b ^ 4 x ^ 4 + 6 a b ^ 5 x ^ 5 + b ^ 6 x ^ 6))

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Maple [A]
time = 0.09, size = 49, normalized size = 0.72

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-5 b^{3} x^{3}+6 a \,b^{2} x^{2}-8 a^{2} b x +16 a^{3}\right )}{35 b^{4}}\)	\(43\)
trager	\(-\frac {2 \sqrt {b x +a}\, \left (-5 b^{3} x^{3}+6 a \,b^{2} x^{2}-8 a^{2} b x +16 a^{3}\right )}{35 b^{4}}\)	\(43\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-5 b^{3} x^{3}+6 a \,b^{2} x^{2}-8 a^{2} b x +16 a^{3}\right )}{35 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {6 a \left (b x +a \right )^{\frac {5}{2}}}{5}+2 a^{2} \left (b x +a \right )^{\frac {3}{2}}-2 a^{3} \sqrt {b x +a}}{b^{4}}\)	\(49\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {6 a \left (b x +a \right )^{\frac {5}{2}}}{5}+2 a^{2} \left (b x +a \right )^{\frac {3}{2}}-2 a^{3} \sqrt {b x +a}}{b^{4}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 56, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.31, size = 42, normalized size = 0.62 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (65) = 130\).
time = 1.28, size = 1640, normalized size = 24.12

result too large to display

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 = 0.00, size = 82, normalized size = 1.21 \begin {gather*} \frac {2 \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.82 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}-\frac {2\,a^3\,\sqrt {a+b\,x}}{b^4}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{3/2}}{b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x)^(1/2),x)

[Out]

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

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