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

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]

4/3*(b^4*c^2-6*a^2*b^2*c+5*a^4)*(a+b*(d*x+c)^(1/2))^(3/2)/b^6/d^3-8/5*a*(-3*b^2*c+5*a^2)*(a+b*(d*x+c)^(1/2))^(
5/2)/b^6/d^3+8/7*(-b^2*c+5*a^2)*(a+b*(d*x+c)^(1/2))^(7/2)/b^6/d^3-20/9*a*(a+b*(d*x+c)^(1/2))^(9/2)/b^6/d^3+4/1
1*(a+b*(d*x+c)^(1/2))^(11/2)/b^6/d^3-4*a*(-b^2*c+a^2)^2*(a+b*(d*x+c)^(1/2))^(1/2)/b^6/d^3

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Rubi [A]  time = 0.16, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \[ \frac {4 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac {4 a \left (a^2-b^2 c\right )^2 \sqrt {a+b \sqrt {c+d x}}}{b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3} \]

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)

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.17, 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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fricas [A]  time = 0.55, size = 140, normalized size = 0.63 \[ -\frac {4 \, {\left (350 \, a b^{4} d^{2} x^{2} + 1472 \, a b^{4} c^{2} - 2688 \, a^{3} b^{2} c + 1280 \, a^{5} - 8 \, {\left (61 \, a b^{4} c - 60 \, a^{3} b^{2}\right )} d x - {\left (315 \, b^{5} d^{2} x^{2} + 480 \, b^{5} c^{2} - 1184 \, a^{2} b^{3} c + 640 \, a^{4} b - 40 \, {\left (9 \, b^{5} c - 10 \, a^{2} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{3465 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*(d*x+c)^(1/2))^(1/2),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 - (3
15*b^5*d^2*x^2 + 480*b^5*c^2 - 1184*a^2*b^3*c + 640*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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giac [A]  time = 0.33, size = 238, normalized size = 1.07 \[ \frac {4 \, {\left (1155 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} c^{2} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} c^{2} - 990 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{2} c + 4158 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{2} c - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{2} c + 6930 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{2} c + 315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a + 4950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} + 5775 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a^{5}\right )}}{3465 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3465*(1155*(sqrt(d*x + c)*b + a)^(3/2)*b^4*c^2 - 3465*sqrt(sqrt(d*x + c)*b + a)*a*b^4*c^2 - 990*(sqrt(d*x +
c)*b + a)^(7/2)*b^2*c + 4158*(sqrt(d*x + c)*b + a)^(5/2)*a*b^2*c - 6930*(sqrt(d*x + c)*b + a)^(3/2)*a^2*b^2*c
+ 6930*sqrt(sqrt(d*x + c)*b + a)*a^3*b^2*c + 315*(sqrt(d*x + c)*b + a)^(11/2) - 1925*(sqrt(d*x + c)*b + a)^(9/
2)*a + 4950*(sqrt(d*x + c)*b + a)^(7/2)*a^2 - 6930*(sqrt(d*x + c)*b + a)^(5/2)*a^3 + 5775*(sqrt(d*x + c)*b + a
)^(3/2)*a^4 - 3465*sqrt(sqrt(d*x + c)*b + a)*a^5)/(b^6*d^3)

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maple [A]  time = 0.00, size = 183, normalized size = 0.82 \[ \frac {-\frac {20 \left (a +\sqrt {d x +c}\, b \right )^{\frac {9}{2}} a}{9}-4 \left (-b^{2} c +a^{2}\right )^{2} \sqrt {a +\sqrt {d x +c}\, b}\, a +\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {11}{2}}}{11}+\frac {4 \left (-2 b^{2} c +10 a^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {7}{2}}}{7}+\frac {4 \left (-4 \left (-b^{2} c +a^{2}\right ) a -\left (-2 b^{2} c +6 a^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}}}{5}+\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a^{2}+\left (-b^{2} c +a^{2}\right )^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}}}{3}}{b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.92, size = 167, normalized size = 0.75 \[ \frac {4 \, {\left (315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a - 990 \, {\left (b^{2} c - 5 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 1386 \, {\left (3 \, a b^{2} c - 5 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3465 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt {\sqrt {d x + c} b + a}\right )}}{3465 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*(d*x+c)^(1/2))^(1/2),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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {a + b \sqrt {c + d x}}}\, dx \]

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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