3.4.43 \(\int \frac {x^4}{(a+b x)^{3/2}} \, dx\) [343]
Optimal. Leaf size=85 \[ -\frac {2 a^4}{b^5 \sqrt {a+b x}}-\frac {8 a^3 \sqrt {a+b x}}{b^5}+\frac {4 a^2 (a+b x)^{3/2}}{b^5}-\frac {8 a (a+b x)^{5/2}}{5 b^5}+\frac {2 (a+b x)^{7/2}}{7 b^5} \]
[Out]
4*a^2*(b*x+a)^(3/2)/b^5-8/5*a*(b*x+a)^(5/2)/b^5+2/7*(b*x+a)^(7/2)/b^5-2*a^4/b^5/(b*x+a)^(1/2)-8*a^3*(b*x+a)^(1
/2)/b^5
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Rubi [A]
time = 0.02, antiderivative size = 85, 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^4}{b^5 \sqrt {a+b x}}-\frac {8 a^3 \sqrt {a+b x}}{b^5}+\frac {4 a^2 (a+b x)^{3/2}}{b^5}-\frac {8 a (a+b x)^{5/2}}{5 b^5}+\frac {2 (a+b x)^{7/2}}{7 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^4/(a + b*x)^(3/2),x]
[Out]
(-2*a^4)/(b^5*Sqrt[a + b*x]) - (8*a^3*Sqrt[a + b*x])/b^5 + (4*a^2*(a + b*x)^(3/2))/b^5 - (8*a*(a + b*x)^(5/2))
/(5*b^5) + (2*(a + b*x)^(7/2))/(7*b^5)
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^4}{(a+b x)^{3/2}} \, dx &=\int \left (\frac {a^4}{b^4 (a+b x)^{3/2}}-\frac {4 a^3}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {4 a (a+b x)^{3/2}}{b^4}+\frac {(a+b x)^{5/2}}{b^4}\right ) \, dx\\ &=-\frac {2 a^4}{b^5 \sqrt {a+b x}}-\frac {8 a^3 \sqrt {a+b x}}{b^5}+\frac {4 a^2 (a+b x)^{3/2}}{b^5}-\frac {8 a (a+b x)^{5/2}}{5 b^5}+\frac {2 (a+b x)^{7/2}}{7 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.67 \begin {gather*} \frac {2 \left (-128 a^4-64 a^3 b x+16 a^2 b^2 x^2-8 a b^3 x^3+5 b^4 x^4\right )}{35 b^5 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^4/(a + b*x)^(3/2),x]
[Out]
(2*(-128*a^4 - 64*a^3*b*x + 16*a^2*b^2*x^2 - 8*a*b^3*x^3 + 5*b^4*x^4))/(35*b^5*Sqrt[a + b*x])
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(516\) vs. \(2(85)=170\).
time = 32.01, size = 484, normalized size = 5.69 \begin {gather*} \frac {2 \sqrt {a} \left (128 a^{13} \left (1-\sqrt {\frac {a+b x}{a}}\right )+64 a^{12} b x \left (20-19 \sqrt {\frac {a+b x}{a}}\right )+16 a^{11} b^2 x^2 \left (360-323 \sqrt {\frac {a+b x}{a}}\right )+40 a^{10} b^3 x^3 \left (384-323 \sqrt {\frac {a+b x}{a}}\right )+5 a^9 b^4 x^4 \left (5376-4199 \sqrt {\frac {a+b x}{a}}\right )+a b^5 x^5 \left (-23091 a^7 \sqrt {\frac {a+b x}{a}}-17292 a^6 b x \sqrt {\frac {a+b x}{a}}-8556 a^5 b^2 x^2 \sqrt {\frac {a+b x}{a}}+128 a^2 b^5 x^5+212 a^2 b^5 x^5 \sqrt {\frac {a+b x}{a}}+124 a b^6 x^6 \sqrt {\frac {a+b x}{a}}+37 b^7 x^7 \sqrt {\frac {a+b x}{a}}\right )+32256 a^8 b^5 x^5+26880 a^7 b^6 x^6+5 b^7 x^7 \left (3072 a^6-498 a^5 b x \sqrt {\frac {a+b x}{a}}+1152 a^5 b x-34 a^4 b^2 x^2 \sqrt {\frac {a+b x}{a}}+256 a^4 b^2 x^2+b^6 x^6 \sqrt {\frac {a+b x}{a}}\right )\right )}{35 b^5 \left (a^{10}+10 a^9 b x+45 a^8 b^2 x^2+120 a^7 b^3 x^3+210 a^6 b^4 x^4+252 a^5 b^5 x^5+210 a^4 b^6 x^6+120 a^3 b^7 x^7+45 a^2 b^8 x^8+10 a b^9 x^9+b^{10} x^{10}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[x^4/(a + b*x)^(3/2),x]')
[Out]
2 Sqrt[a] (128 a ^ 13 (1 - Sqrt[(a + b x) / a]) + 64 a ^ 12 b x (20 - 19 Sqrt[(a + b x) / a]) + 16 a ^ 11 b ^
2 x ^ 2 (360 - 323 Sqrt[(a + b x) / a]) + 40 a ^ 10 b ^ 3 x ^ 3 (384 - 323 Sqrt[(a + b x) / a]) + 5 a ^ 9 b ^
4 x ^ 4 (5376 - 4199 Sqrt[(a + b x) / a]) + a b ^ 5 x ^ 5 (-23091 a ^ 7 Sqrt[(a + b x) / a] - 17292 a ^ 6 b x
Sqrt[(a + b x) / a] - 8556 a ^ 5 b ^ 2 x ^ 2 Sqrt[(a + b x) / a] + 128 a ^ 2 b ^ 5 x ^ 5 + 212 a ^ 2 b ^ 5 x ^
5 Sqrt[(a + b x) / a] + 124 a b ^ 6 x ^ 6 Sqrt[(a + b x) / a] + 37 b ^ 7 x ^ 7 Sqrt[(a + b x) / a]) + 32256 a
^ 8 b ^ 5 x ^ 5 + 26880 a ^ 7 b ^ 6 x ^ 6 + 5 b ^ 7 x ^ 7 (3072 a ^ 6 - 498 a ^ 5 b x Sqrt[(a + b x) / a] + 1
152 a ^ 5 b x - 34 a ^ 4 b ^ 2 x ^ 2 Sqrt[(a + b x) / a] + 256 a ^ 4 b ^ 2 x ^ 2 + b ^ 6 x ^ 6 Sqrt[(a + b x)
/ a])) / (35 b ^ 5 (a ^ 10 + 10 a ^ 9 b x + 45 a ^ 8 b ^ 2 x ^ 2 + 120 a ^ 7 b ^ 3 x ^ 3 + 210 a ^ 6 b ^ 4 x ^
4 + 252 a ^ 5 b ^ 5 x ^ 5 + 210 a ^ 4 b ^ 6 x ^ 6 + 120 a ^ 3 b ^ 7 x ^ 7 + 45 a ^ 2 b ^ 8 x ^ 8 + 10 a b ^ 9
x ^ 9 + b ^ 10 x ^ 10))
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Maple [A]
time = 0.09, size = 62, normalized size = 0.73
| | |
| method |
result |
size |
| | |
| gosper |
\(-\frac {2 \left (-5 b^{4} x^{4}+8 a \,b^{3} x^{3}-16 a^{2} b^{2} x^{2}+64 a^{3} b x +128 a^{4}\right )}{35 \sqrt {b x +a}\, b^{5}}\) |
\(54\) |
| trager |
\(-\frac {2 \left (-5 b^{4} x^{4}+8 a \,b^{3} x^{3}-16 a^{2} b^{2} x^{2}+64 a^{3} b x +128 a^{4}\right )}{35 \sqrt {b x +a}\, b^{5}}\) |
\(54\) |
| risch |
\(-\frac {2 \left (-5 b^{3} x^{3}+13 a \,b^{2} x^{2}-29 a^{2} b x +93 a^{3}\right ) \sqrt {b x +a}}{35 b^{5}}-\frac {2 a^{4}}{b^{5} \sqrt {b x +a}}\) |
\(59\) |
| derivativedivides |
\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {8 a \left (b x +a \right )^{\frac {5}{2}}}{5}+4 a^{2} \left (b x +a \right )^{\frac {3}{2}}-8 a^{3} \sqrt {b x +a}-\frac {2 a^{4}}{\sqrt {b x +a}}}{b^{5}}\) |
\(62\) |
| default |
\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {8 a \left (b x +a \right )^{\frac {5}{2}}}{5}+4 a^{2} \left (b x +a \right )^{\frac {3}{2}}-8 a^{3} \sqrt {b x +a}-\frac {2 a^{4}}{\sqrt {b x +a}}}{b^{5}}\) |
\(62\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
[Out]
2/b^5*(1/7*(b*x+a)^(7/2)-4/5*a*(b*x+a)^(5/2)+2*a^2*(b*x+a)^(3/2)-4*a^3*(b*x+a)^(1/2)-a^4/(b*x+a)^(1/2))
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Maxima [A]
time = 0.26, size = 71, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{5}} - \frac {8 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{5}} + \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{b^{5}} - \frac {8 \, \sqrt {b x + a} a^{3}}{b^{5}} - \frac {2 \, a^{4}}{\sqrt {b x + a} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(3/2),x, algorithm="maxima")
[Out]
2/7*(b*x + a)^(7/2)/b^5 - 8/5*(b*x + a)^(5/2)*a/b^5 + 4*(b*x + a)^(3/2)*a^2/b^5 - 8*sqrt(b*x + a)*a^3/b^5 - 2*
a^4/(sqrt(b*x + a)*b^5)
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Fricas [A]
time = 0.30, size = 63, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (5 \, b^{4} x^{4} - 8 \, a b^{3} x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a^{3} b x - 128 \, a^{4}\right )} \sqrt {b x + a}}{35 \, {\left (b^{6} x + a b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(3/2),x, algorithm="fricas")
[Out]
2/35*(5*b^4*x^4 - 8*a*b^3*x^3 + 16*a^2*b^2*x^2 - 64*a^3*b*x - 128*a^4)*sqrt(b*x + a)/(b^6*x + a*b^5)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3606 vs.
\(2 (82) = 164\).
time = 2.20, size = 3606, normalized size = 42.42
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4/(b*x+a)**(3/2),x)
[Out]
-256*a**(87/2)*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3
+ 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b
**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 256*a**(87/2)/(35*a**40*b**5 + 350*a**39*b**6*x + 1
575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 2432*a**(
85/2)*b*x*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 73
50*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*
x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2560*a**(85/2)*b*x/(35*a**40*b**5 + 350*a**39*b**6*x + 1
575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 10336*a**
(83/2)*b**2*x**2*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x*
*3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32
*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 11520*a**(83/2)*b**2*x**2/(35*a**40*b**5 + 350*a*
*39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350
*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**1
0) - 25840*a**(81/2)*b**3*x**3*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200
*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**
7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 30720*a**(81/2)*b**3*x**3/(35*a**40
*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**
10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a*
*30*b**15*x**10) - 41990*a**(79/2)*b**4*x**4*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b*
*7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a
**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 53760*a**(79/2)*b**4*
x**4/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 +
8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**1
4*x**9 + 35*a**30*b**15*x**10) - 46182*a**(77/2)*b**5*x**5*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x +
1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11
*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 64512*a
**(77/2)*b**5*x**5/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**3
6*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 +
350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 34584*a**(75/2)*b**6*x**6*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*
a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 73
50*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x*
*10) + 53760*a**(75/2)*b**6*x**6/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x*
*3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32
*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 17112*a**(73/2)*b**7*x**7*sqrt(1 + b*x/a)/(35*a**
40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b
**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*
a**30*b**15*x**10) + 30720*a**(73/2)*b**7*x**7/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200
*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**
7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 4980*a**(71/2)*b**8*x**8*sqrt(1 + b
*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 +
8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**
14*x**9 + 35*a**30*b**15*x**10) + 11520*a**(71/2)*b**8*x**8/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**
7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a*
*33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 340*a**(69/2)*b**9*x**
9*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36
*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 3
50*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2560*a**(69/2)*b**9*x**9/(35*a**40*b**5 + 350*a**39*b**6*x + 157
5*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**
6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 424*a**(67/
2)*b**10*x**10*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3
+ 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b
**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 256*a**(67/2)*b**10*x**10/(35*a**40*b**5 + 350*a**3
9*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a
**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10)
+ 248*a**(65/2)*b**11*x**11*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a
**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7
+ 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 74*a**(63/2)*b**12*x**12*sqrt(1 + b*x
/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8
820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14
*x**9 + 35*a**30*b**15*x**10) + 10*a**(61/2)*b**13*x**13*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1
575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10)
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Giac [A]
time = 0.00, size = 115, normalized size = 1.35 \begin {gather*} 2 \left (\frac {\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3} b^{30}-\frac {4}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a b^{30}+2 \sqrt {a+b x} \left (a+b x\right ) a^{2} b^{30}-4 \sqrt {a+b x} a^{3} b^{30}}{b^{35}}-\frac {a^{4}}{b^{5} \sqrt {a+b x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(3/2),x)
[Out]
-2*a^4/(sqrt(b*x + a)*b^5) + 2/35*(5*(b*x + a)^(7/2)*b^30 - 28*(b*x + a)^(5/2)*a*b^30 + 70*(b*x + a)^(3/2)*a^2
*b^30 - 140*sqrt(b*x + a)*a^3*b^30)/b^35
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Mupad [B]
time = 0.03, size = 71, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {8\,a^3\,\sqrt {a+b\,x}}{b^5}+\frac {4\,a^2\,{\left (a+b\,x\right )}^{3/2}}{b^5}-\frac {2\,a^4}{b^5\,\sqrt {a+b\,x}}-\frac {8\,a\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(a + b*x)^(3/2),x)
[Out]
(2*(a + b*x)^(7/2))/(7*b^5) - (8*a^3*(a + b*x)^(1/2))/b^5 + (4*a^2*(a + b*x)^(3/2))/b^5 - (2*a^4)/(b^5*(a + b*
x)^(1/2)) - (8*a*(a + b*x)^(5/2))/(5*b^5)
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