3.465 \(\int x^2 \sqrt{a+b \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=224 \[ \frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3} \]

[Out]

(-4*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^6*d^3) + (4*(5*a^4 - 6*a
^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^6*d^3) - (8*a*(5*a^2 - 3*b
^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^6*d^3) + (8*(5*a^2 - b^2*c)*(a + b*Sqrt[
c + d*x])^(9/2))/(9*b^6*d^3) - (20*a*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^6*d^3)
+ (4*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^6*d^3)

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Rubi [A]  time = 0.371229, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^6*d^3) + (4*(5*a^4 - 6*a
^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^6*d^3) - (8*a*(5*a^2 - 3*b
^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^6*d^3) + (8*(5*a^2 - b^2*c)*(a + b*Sqrt[
c + d*x])^(9/2))/(9*b^6*d^3) - (20*a*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^6*d^3)
+ (4*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^6*d^3)

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Rubi in Sympy [A]  time = 22.5591, size = 211, normalized size = 0.94 \[ - \frac{20 a \left (a + b \sqrt{c + d x}\right )^{\frac{11}{2}}}{11 b^{6} d^{3}} - \frac{8 a \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}} \left (5 a^{2} - 3 b^{2} c\right )}{7 b^{6} d^{3}} - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{2}}{3 b^{6} d^{3}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{13}{2}}}{13 b^{6} d^{3}} + \frac{8 \left (a + b \sqrt{c + d x}\right )^{\frac{9}{2}} \left (5 a^{2} - b^{2} c\right )}{9 b^{6} d^{3}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}} \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{5 b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*a*(a + b*sqrt(c + d*x))**(11/2)/(11*b**6*d**3) - 8*a*(a + b*sqrt(c + d*x))**
(7/2)*(5*a**2 - 3*b**2*c)/(7*b**6*d**3) - 4*a*(a + b*sqrt(c + d*x))**(3/2)*(a**2
- b**2*c)**2/(3*b**6*d**3) + 4*(a + b*sqrt(c + d*x))**(13/2)/(13*b**6*d**3) + 8
*(a + b*sqrt(c + d*x))**(9/2)*(5*a**2 - b**2*c)/(9*b**6*d**3) + 4*(a + b*sqrt(c
+ d*x))**(5/2)*(5*a**4 - 6*a**2*b**2*c + b**4*c**2)/(5*b**6*d**3)

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Mathematica [A]  time = 0.183388, size = 147, normalized size = 0.66 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2} \left (-1280 a^5+1920 a^4 b \sqrt{c+d x}+32 a^3 b^2 (68 c-75 d x)+16 a^2 b^3 \sqrt{c+d x} (175 d x-254 c)-6 a b^4 \left (96 c^2-380 c d x+525 d^2 x^2\right )+77 b^5 \sqrt{c+d x} \left (32 c^2-40 c d x+45 d^2 x^2\right )\right )}{45045 b^6 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(4*(a + b*Sqrt[c + d*x])^(3/2)*(-1280*a^5 + 32*a^3*b^2*(68*c - 75*d*x) + 1920*a^
4*b*Sqrt[c + d*x] + 16*a^2*b^3*Sqrt[c + d*x]*(-254*c + 175*d*x) + 77*b^5*Sqrt[c
+ d*x]*(32*c^2 - 40*c*d*x + 45*d^2*x^2) - 6*a*b^4*(96*c^2 - 380*c*d*x + 525*d^2*
x^2)))/(45045*b^6*d^3)

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Maple [A]  time = 0.003, size = 183, normalized size = 0.8 \[ 4\,{\frac{1}{{d}^{3}{b}^{6}} \left ( 1/13\, \left ( a+b\sqrt{dx+c} \right ) ^{13/2}-{\frac{5\,a \left ( a+b\sqrt{dx+c} \right ) ^{11/2}}{11}}+1/9\, \left ( -2\,{b}^{2}c+10\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{9/2}+1/7\, \left ( -4\, \left ( -{b}^{2}c+{a}^{2} \right ) a-a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) \right ) \left ( a+b\sqrt{dx+c} \right ) ^{7/2}+1/5\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}+4\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}a \left ( a+b\sqrt{dx+c} \right ) ^{3/2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*(d*x+c)^(1/2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.725133, size = 225, normalized size = 1. \[ \frac{4 \,{\left (3465 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{13}{2}} - 20475 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} a - 10010 \,{\left (b^{2} c - 5 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} + 12870 \,{\left (3 \, a b^{2} c - 5 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} + 9009 \,{\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} - 15015 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}\right )}}{45045 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)*x^2,x, algorithm="maxima")

[Out]

4/45045*(3465*(sqrt(d*x + c)*b + a)^(13/2) - 20475*(sqrt(d*x + c)*b + a)^(11/2)*
a - 10010*(b^2*c - 5*a^2)*(sqrt(d*x + c)*b + a)^(9/2) + 12870*(3*a*b^2*c - 5*a^3
)*(sqrt(d*x + c)*b + a)^(7/2) + 9009*(b^4*c^2 - 6*a^2*b^2*c + 5*a^4)*(sqrt(d*x +
c)*b + a)^(5/2) - 15015*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*(sqrt(d*x + c)*b + a)^(
3/2))/(b^6*d^3)

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Fricas [A]  time = 0.358349, size = 248, normalized size = 1.11 \[ \frac{4 \,{\left (3465 \, b^{6} d^{3} x^{3} + 2464 \, b^{6} c^{3} - 4640 \, a^{2} b^{4} c^{2} + 4096 \, a^{4} b^{2} c - 1280 \, a^{6} + 35 \,{\left (11 \, b^{6} c - 10 \, a^{2} b^{4}\right )} d^{2} x^{2} - 8 \,{\left (77 \, b^{6} c^{2} - 127 \, a^{2} b^{4} c + 60 \, a^{4} b^{2}\right )} d x +{\left (315 \, a b^{5} d^{2} x^{2} + 1888 \, a b^{5} c^{2} - 1888 \, a^{3} b^{3} c + 640 \, a^{5} b - 400 \,{\left (2 \, a b^{5} c - a^{3} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{45045 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)*x^2,x, algorithm="fricas")

[Out]

4/45045*(3465*b^6*d^3*x^3 + 2464*b^6*c^3 - 4640*a^2*b^4*c^2 + 4096*a^4*b^2*c - 1
280*a^6 + 35*(11*b^6*c - 10*a^2*b^4)*d^2*x^2 - 8*(77*b^6*c^2 - 127*a^2*b^4*c + 6
0*a^4*b^2)*d*x + (315*a*b^5*d^2*x^2 + 1888*a*b^5*c^2 - 1888*a^3*b^3*c + 640*a^5*
b - 400*(2*a*b^5*c - a^3*b^3)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^6
*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \sqrt{a + b \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(a + b*sqrt(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.335948, size = 923, normalized size = 4.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)*x^2,x, algorithm="giac")

[Out]

4/45045*(9009*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*b^6*c^2*si
gn((sqrt(d*x + c)*b + a)*b - a*b) - 15015*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(
d*x + c)*b + a)*a*b^6*c^2*sign((sqrt(d*x + c)*b + a)*b - a*b) - 10010*sqrt((sqrt
(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*b^4*c*sign((sqrt(d*x + c)*b + a)*b
- a*b) + 38610*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a*b^4*c*
sign((sqrt(d*x + c)*b + a)*b - a*b) - 54054*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqr
t(d*x + c)*b + a)^2*a^2*b^4*c*sign((sqrt(d*x + c)*b + a)*b - a*b) + 30030*sqrt((
sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^3*b^4*c*sign((sqrt(d*x + c)*b
+ a)*b - a*b) + 3465*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^6*b^2
*sign((sqrt(d*x + c)*b + a)*b - a*b) - 20475*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sq
rt(d*x + c)*b + a)^5*a*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b) + 50050*sqrt((sqr
t(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a^2*b^2*sign((sqrt(d*x + c)*b + a
)*b - a*b) - 64350*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^3*b
^2*sign((sqrt(d*x + c)*b + a)*b - a*b) + 45045*sqrt((sqrt(d*x + c)*b + a)*b^2)*(
sqrt(d*x + c)*b + a)^2*a^4*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b) - 15015*sqrt(
(sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^5*b^2*sign((sqrt(d*x + c)*b +
a)*b - a*b))*abs(b)/(b^10*d^3)