3.2233 \(\int \sqrt{a+b \sqrt{x}} x^2 \, dx\)

Optimal. Leaf size=132 \[ \frac{40 a^2 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^6}-\frac{40 a^3 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^6}+\frac{4 a^4 \left (a+b \sqrt{x}\right )^{5/2}}{b^6}-\frac{4 a^5 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^6}+\frac{4 \left (a+b \sqrt{x}\right )^{13/2}}{13 b^6}-\frac{20 a \left (a+b \sqrt{x}\right )^{11/2}}{11 b^6} \]

[Out]

(-4*a^5*(a + b*Sqrt[x])^(3/2))/(3*b^6) + (4*a^4*(a + b*Sqrt[x])^(5/2))/b^6 - (40*a^3*(a + b*Sqrt[x])^(7/2))/(7
*b^6) + (40*a^2*(a + b*Sqrt[x])^(9/2))/(9*b^6) - (20*a*(a + b*Sqrt[x])^(11/2))/(11*b^6) + (4*(a + b*Sqrt[x])^(
13/2))/(13*b^6)

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Rubi [A]  time = 0.0618827, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{40 a^2 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^6}-\frac{40 a^3 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^6}+\frac{4 a^4 \left (a+b \sqrt{x}\right )^{5/2}}{b^6}-\frac{4 a^5 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^6}+\frac{4 \left (a+b \sqrt{x}\right )^{13/2}}{13 b^6}-\frac{20 a \left (a+b \sqrt{x}\right )^{11/2}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[x]]*x^2,x]

[Out]

(-4*a^5*(a + b*Sqrt[x])^(3/2))/(3*b^6) + (4*a^4*(a + b*Sqrt[x])^(5/2))/b^6 - (40*a^3*(a + b*Sqrt[x])^(7/2))/(7
*b^6) + (40*a^2*(a + b*Sqrt[x])^(9/2))/(9*b^6) - (20*a*(a + b*Sqrt[x])^(11/2))/(11*b^6) + (4*(a + b*Sqrt[x])^(
13/2))/(13*b^6)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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{a+b \sqrt{x}} x^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 \sqrt{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^5 \sqrt{a+b x}}{b^5}+\frac{5 a^4 (a+b x)^{3/2}}{b^5}-\frac{10 a^3 (a+b x)^{5/2}}{b^5}+\frac{10 a^2 (a+b x)^{7/2}}{b^5}-\frac{5 a (a+b x)^{9/2}}{b^5}+\frac{(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a^5 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^6}+\frac{4 a^4 \left (a+b \sqrt{x}\right )^{5/2}}{b^6}-\frac{40 a^3 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^6}+\frac{40 a^2 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^6}-\frac{20 a \left (a+b \sqrt{x}\right )^{11/2}}{11 b^6}+\frac{4 \left (a+b \sqrt{x}\right )^{13/2}}{13 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0378882, size = 78, normalized size = 0.59 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2} \left (560 a^2 b^3 x^{3/2}-480 a^3 b^2 x+384 a^4 b \sqrt{x}-256 a^5-630 a b^4 x^2+693 b^5 x^{5/2}\right )}{9009 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[x]]*x^2,x]

[Out]

(4*(a + b*Sqrt[x])^(3/2)*(-256*a^5 + 384*a^4*b*Sqrt[x] - 480*a^3*b^2*x + 560*a^2*b^3*x^(3/2) - 630*a*b^4*x^2 +
693*b^5*x^(5/2)))/(9009*b^6)

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Maple [A]  time = 0.005, size = 85, normalized size = 0.6 \begin{align*} 4\,{\frac{1}{{b}^{6}} \left ( 1/13\, \left ( a+b\sqrt{x} \right ) ^{13/2}-{\frac{5\,a \left ( a+b\sqrt{x} \right ) ^{11/2}}{11}}+{\frac{10\,{a}^{2} \left ( a+b\sqrt{x} \right ) ^{9/2}}{9}}-{\frac{10\,{a}^{3} \left ( a+b\sqrt{x} \right ) ^{7/2}}{7}}+{a}^{4} \left ( a+b\sqrt{x} \right ) ^{5/2}-1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}{a}^{5} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/b^6*(1/13*(a+b*x^(1/2))^(13/2)-5/11*a*(a+b*x^(1/2))^(11/2)+10/9*a^2*(a+b*x^(1/2))^(9/2)-10/7*a^3*(a+b*x^(1/2
))^(7/2)+a^4*(a+b*x^(1/2))^(5/2)-1/3*(a+b*x^(1/2))^(3/2)*a^5)

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Maxima [A]  time = 0.9642, size = 132, normalized size = 1. \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{13}{2}}}{13 \, b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )}^{\frac{11}{2}} a}{11 \, b^{6}} + \frac{40 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} a^{2}}{9 \, b^{6}} - \frac{40 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{6}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{5}}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/13*(b*sqrt(x) + a)^(13/2)/b^6 - 20/11*(b*sqrt(x) + a)^(11/2)*a/b^6 + 40/9*(b*sqrt(x) + a)^(9/2)*a^2/b^6 - 40
/7*(b*sqrt(x) + a)^(7/2)*a^3/b^6 + 4*(b*sqrt(x) + a)^(5/2)*a^4/b^6 - 4/3*(b*sqrt(x) + a)^(3/2)*a^5/b^6

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Fricas [A]  time = 1.3652, size = 189, normalized size = 1.43 \begin{align*} \frac{4 \,{\left (693 \, b^{6} x^{3} - 70 \, a^{2} b^{4} x^{2} - 96 \, a^{4} b^{2} x - 256 \, a^{6} +{\left (63 \, a b^{5} x^{2} + 80 \, a^{3} b^{3} x + 128 \, a^{5} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{9009 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9009*(693*b^6*x^3 - 70*a^2*b^4*x^2 - 96*a^4*b^2*x - 256*a^6 + (63*a*b^5*x^2 + 80*a^3*b^3*x + 128*a^5*b)*sqrt
(x))*sqrt(b*sqrt(x) + a)/b^6

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Sympy [B]  time = 11.626, size = 8588, normalized size = 65.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1024*a**(153/2)*x**18*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**
68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 4
5090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15
*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 1024*a**(153/2)*x**18/(9009*a
**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297
285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**
(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a
**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9
009*a**55*b**21*x**(51/2)) - 14848*a**(151/2)*b*x**(37/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 13513
5*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 +
27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62
*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) +
4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/
2)) + 15360*a**(151/2)*b*x**(37/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x*
*19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a*
*64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2)
+ 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**1
9*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 100224*a**(149/2)*b**2*x**19*sqrt(1 + b
*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**
9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972
915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16
*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a*
*56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 107520*a**(149/2)*b**2*x**19/(9009*a**70*b**6*x**18 + 135135*a
**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27
054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b*
*14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 409
9095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2))
- 417600*a**(147/2)*b**3*x**(39/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2)
+ 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*
x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 4509004
5*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x*
*24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 465920*a**(147/2
)*b**3*x**(39/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**6
7*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 +
57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*
b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 1351
35*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 1200600*a**(145/2)*b**4*x**20*sqrt(1 + b*sqrt(x)/a)/(9009
*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 122
97285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x
**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285
*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 +
9009*a**55*b**21*x**(51/2)) + 1397760*a**(145/2)*b**4*x**20/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37
/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**
11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 4509
0045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18
*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 2521260*a**(1
43/2)*b**5*x**(41/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68
*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 450
90045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x
**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a*
*57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 3075072*a**(143/2)*b**5*x**(41/
2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/
2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63
*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 +
12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20
*x**25 + 9009*a**55*b**21*x**(51/2)) - 3988992*a**(141/2)*b**6*x**21*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x*
*18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b*
*10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 579
72915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x
**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b*
*21*x**(51/2)) + 5125120*a**(141/2)*b**6*x**21/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a
**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) +
45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**
15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 94594
5*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 4802592*a**(139/2)*b**7*x**
(43/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 +
4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b*
*12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 270
54027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(
49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 6589440*a**(139/2)*b**7*x**(43/2)/(9009*a**70
*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*
a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/
2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59
*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*
a**55*b**21*x**(51/2)) - 4232160*a**(137/2)*b**8*x**22*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a
**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27
054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b*
*14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 409
9095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2))
+ 6589440*a**(137/2)*b**8*x**22/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**1
9 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**6
4*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) +
27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*
x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 1935440*a**(135/2)*b**9*x**(45/2)*sqrt(1
+ b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*
b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57
972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b*
*16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135
*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 5125120*a**(135/2)*b**9*x**(45/2)/(9009*a**70*b**6*x**18 +
135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x*
*20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*
a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/
2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x*
*(51/2)) + 2214784*a**(133/2)*b**10*x**23*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**
(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*
b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 4
5090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b*
*18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 3075072*a*
*(133/2)*b**10*x**23/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*
a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**2
1 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a*
*60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) +
135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 8060832*a**(131/2)*b**11*x**(47/2)*sqrt(1 + b*sqrt(x)
/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39
/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**6
3*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 +
12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**2
0*x**25 + 9009*a**55*b**21*x**(51/2)) + 1397760*a**(131/2)*b**11*x**(47/2)/(9009*a**70*b**6*x**18 + 135135*a**
69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 2705
4027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**1
4*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 40990
95*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
14375088*a**(129/2)*b**12*x**24*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) +
945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**
(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a
**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24
+ 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 465920*a**(129/2)*b
**12*x**24/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9
*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 579729
15*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*
x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**
56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 18620280*a**(127/2)*b**13*x**(49/2)*sqrt(1 + b*sqrt(x)/a)/(9009
*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 122
97285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x
**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285
*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 +
9009*a**55*b**21*x**(51/2)) + 107520*a**(127/2)*b**13*x**(49/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x*
*(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65
*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 +
45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b
**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 18558720*
a**(125/2)*b**14*x**25*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**
68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 4
5090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15
*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 15360*a**(125/2)*b**14*x**25/
(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2)
+ 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b*
*13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 122
97285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x*
*25 + 9009*a**55*b**21*x**(51/2)) + 14360352*a**(123/2)*b**15*x**(51/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6
*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66
*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**1
7*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55
*b**21*x**(51/2)) + 1024*a**(123/2)*b**15*x**(51/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945
945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41
/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**6
1*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 +
945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 8569248*a**(121/2)*b**
16*x**26*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19
+ 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*
b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 2
7054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x*
*(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 3873456*a**(119/2)*b**17*x**(53/2)*sqrt(1 +
b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b
**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 579
72915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**
16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*
a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 1283840*a**(117/2)*b**18*x**27*sqrt(1 + b*sqrt(x)/a)/(9009*a
**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297
285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**
(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a
**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9
009*a**55*b**21*x**(51/2)) + 294560*a**(115/2)*b**19*x**(55/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 +
135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x*
*20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*
a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/
2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x*
*(51/2)) + 41832*a**(113/2)*b**20*x**28*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(3
7/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b*
*11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 450
90045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**1
8*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 2772*a**(111
/2)*b**21*x**(57/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*
b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 4509
0045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x*
*(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**
57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2))

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Giac [A]  time = 1.14483, size = 115, normalized size = 0.87 \begin{align*} \frac{4 \,{\left (693 \,{\left (b \sqrt{x} + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b \sqrt{x} + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{5}\right )}}{9009 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9009*(693*(b*sqrt(x) + a)^(13/2) - 4095*(b*sqrt(x) + a)^(11/2)*a + 10010*(b*sqrt(x) + a)^(9/2)*a^2 - 12870*(
b*sqrt(x) + a)^(7/2)*a^3 + 9009*(b*sqrt(x) + a)^(5/2)*a^4 - 3003*(b*sqrt(x) + a)^(3/2)*a^5)/b^6