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

Optimal. Leaf size=222 \[ \frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \sqrt{a+b \sqrt{c+d x}}}{b^6 d^3}+\frac{4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3} \]

[Out]

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

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Rubi [A]  time = 0.3718, antiderivative size = 222, 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 \[ \frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \sqrt{a+b \sqrt{c+d x}}}{b^6 d^3}+\frac{4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3} \]

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

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Rubi in Sympy [A]  time = 22.3999, size = 209, normalized size = 0.94 \[ - \frac{20 a \left (a + b \sqrt{c + d x}\right )^{\frac{9}{2}}}{9 b^{6} d^{3}} - \frac{8 a \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}} \left (5 a^{2} - 3 b^{2} c\right )}{5 b^{6} d^{3}} - \frac{4 a \sqrt{a + b \sqrt{c + d x}} \left (a^{2} - b^{2} c\right )^{2}}{b^{6} d^{3}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{11}{2}}}{11 b^{6} d^{3}} + \frac{8 \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}} \left (5 a^{2} - b^{2} c\right )}{7 b^{6} d^{3}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{3 b^{6} d^{3}} \]

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

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Mathematica [A]  time = 0.347656, size = 147, normalized size = 0.66 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-1280 a^5+640 a^4 b \sqrt{c+d x}+96 a^3 b^2 (28 c-5 d x)-16 a^2 b^3 (74 c-25 d x) \sqrt{c+d x}-2 a b^4 \left (736 c^2-244 c d x+175 d^2 x^2\right )+15 b^5 \sqrt{c+d x} \left (32 c^2-24 c d x+21 d^2 x^2\right )\right )}{3465 b^6 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-1280*a^5 + 96*a^3*b^2*(28*c - 5*d*x) + 640*a^4*b*
Sqrt[c + d*x] - 16*a^2*b^3*(74*c - 25*d*x)*Sqrt[c + d*x] + 15*b^5*Sqrt[c + d*x]*
(32*c^2 - 24*c*d*x + 21*d^2*x^2) - 2*a*b^4*(736*c^2 - 244*c*d*x + 175*d^2*x^2)))
/(3465*b^6*d^3)

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

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

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Maxima [A]  time = 0.703136, size = 225, normalized size = 1.01 \[ \frac{4 \,{\left (315 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} - 1925 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} a - 990 \,{\left (b^{2} c - 5 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} + 1386 \,{\left (3 \, a b^{2} c - 5 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} - 3465 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt{\sqrt{d x + c} b + a}\right )}}{3465 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3465*(315*(sqrt(d*x + c)*b + a)^(11/2) - 1925*(sqrt(d*x + c)*b + a)^(9/2)*a -
990*(b^2*c - 5*a^2)*(sqrt(d*x + c)*b + a)^(7/2) + 1386*(3*a*b^2*c - 5*a^3)*(sqrt
(d*x + c)*b + a)^(5/2) + 1155*(b^4*c^2 - 6*a^2*b^2*c + 5*a^4)*(sqrt(d*x + c)*b +
a)^(3/2) - 3465*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*sqrt(sqrt(d*x + c)*b + a))/(b^6
*d^3)

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Fricas [A]  time = 0.335953, size = 189, normalized size = 0.85 \[ -\frac{4 \,{\left (350 \, a b^{4} d^{2} x^{2} + 1472 \, a b^{4} c^{2} - 2688 \, a^{3} b^{2} c + 1280 \, a^{5} - 8 \,{\left (61 \, a b^{4} c - 60 \, a^{3} b^{2}\right )} d x -{\left (315 \, b^{5} d^{2} x^{2} + 480 \, b^{5} c^{2} - 1184 \, a^{2} b^{3} c + 640 \, a^{4} b - 40 \,{\left (9 \, b^{5} c - 10 \, a^{2} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{3465 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3465*(350*a*b^4*d^2*x^2 + 1472*a*b^4*c^2 - 2688*a^3*b^2*c + 1280*a^5 - 8*(61*
a*b^4*c - 60*a^3*b^2)*d*x - (315*b^5*d^2*x^2 + 480*b^5*c^2 - 1184*a^2*b^3*c + 64
0*a^4*b - 40*(9*b^5*c - 10*a^2*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. \[ \int \frac{x^{2}}{\sqrt{a + b \sqrt{c + d x}}}\, dx \]

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 = 0.31854, size = 849, normalized size = 3.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3465*(1155*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*b^4*c^2*sign(
(sqrt(d*x + c)*b + a)*b - a*b) - 3465*sqrt((sqrt(d*x + c)*b + a)*b^2)*a*b^4*c^2*
sign((sqrt(d*x + c)*b + a)*b - a*b) - 990*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(
d*x + c)*b + a)^3*b^2*c*sign((sqrt(d*x + c)*b + a)*b - a*b) + 4158*sqrt((sqrt(d*
x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a*b^2*c*sign((sqrt(d*x + c)*b + a)*b
- a*b) - 6930*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^2*b^2*c*si
gn((sqrt(d*x + c)*b + a)*b - a*b) + 6930*sqrt((sqrt(d*x + c)*b + a)*b^2)*a^3*b^2
*c*sign((sqrt(d*x + c)*b + a)*b - a*b) + 315*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sq
rt(d*x + c)*b + a)^5*sign((sqrt(d*x + c)*b + a)*b - a*b) - 1925*sqrt((sqrt(d*x +
c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a*sign((sqrt(d*x + c)*b + a)*b - a*b) +
4950*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^2*sign((sqrt(d*x
+ c)*b + a)*b - a*b) - 6930*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a
)^2*a^3*sign((sqrt(d*x + c)*b + a)*b - a*b) + 5775*sqrt((sqrt(d*x + c)*b + a)*b^
2)*(sqrt(d*x + c)*b + a)*a^4*sign((sqrt(d*x + c)*b + a)*b - a*b) - 3465*sqrt((sq
rt(d*x + c)*b + a)*b^2)*a^5*sign((sqrt(d*x + c)*b + a)*b - a*b))/(b^6*d^3*abs(b)
)