3.681 \(\int \sqrt{d x} (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)

Optimal. Leaf size=131 \[ \frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]

[Out]

(2*a^6*(d*x)^(3/2))/(3*d) + (12*a^5*b*(d*x)^(7/2))/(7*d^3) + (30*a^4*b^2*(d*x)^(11/2))/(11*d^5) + (8*a^3*b^3*(
d*x)^(15/2))/(3*d^7) + (30*a^2*b^4*(d*x)^(19/2))/(19*d^9) + (12*a*b^5*(d*x)^(23/2))/(23*d^11) + (2*b^6*(d*x)^(
27/2))/(27*d^13)

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Rubi [A]  time = 0.0610318, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {28, 270} \[ \frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*a^6*(d*x)^(3/2))/(3*d) + (12*a^5*b*(d*x)^(7/2))/(7*d^3) + (30*a^4*b^2*(d*x)^(11/2))/(11*d^5) + (8*a^3*b^3*(
d*x)^(15/2))/(3*d^7) + (30*a^2*b^4*(d*x)^(19/2))/(19*d^9) + (12*a*b^5*(d*x)^(23/2))/(23*d^11) + (2*b^6*(d*x)^(
27/2))/(27*d^13)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sqrt{d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int \sqrt{d x} \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 \sqrt{d x}+\frac{6 a^5 b^7 (d x)^{5/2}}{d^2}+\frac{15 a^4 b^8 (d x)^{9/2}}{d^4}+\frac{20 a^3 b^9 (d x)^{13/2}}{d^6}+\frac{15 a^2 b^{10} (d x)^{17/2}}{d^8}+\frac{6 a b^{11} (d x)^{21/2}}{d^{10}}+\frac{b^{12} (d x)^{25/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0194856, size = 77, normalized size = 0.59 \[ \frac{2 x \sqrt{d x} \left (717255 a^2 b^4 x^8+1211364 a^3 b^3 x^6+1238895 a^4 b^2 x^4+778734 a^5 b x^2+302841 a^6+237006 a b^5 x^{10}+33649 b^6 x^{12}\right )}{908523} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*x*Sqrt[d*x]*(302841*a^6 + 778734*a^5*b*x^2 + 1238895*a^4*b^2*x^4 + 1211364*a^3*b^3*x^6 + 717255*a^2*b^4*x^8
+ 237006*a*b^5*x^10 + 33649*b^6*x^12))/908523

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Maple [A]  time = 0.05, size = 74, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 33649\,{b}^{6}{x}^{12}+237006\,a{b}^{5}{x}^{10}+717255\,{a}^{2}{b}^{4}{x}^{8}+1211364\,{a}^{3}{b}^{3}{x}^{6}+1238895\,{a}^{4}{b}^{2}{x}^{4}+778734\,{a}^{5}b{x}^{2}+302841\,{a}^{6} \right ) }{908523}\sqrt{dx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x)

[Out]

2/908523*x*(33649*b^6*x^12+237006*a*b^5*x^10+717255*a^2*b^4*x^8+1211364*a^3*b^3*x^6+1238895*a^4*b^2*x^4+778734
*a^5*b*x^2+302841*a^6)*(d*x)^(1/2)

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Maxima [A]  time = 0.966889, size = 142, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (33649 \, \left (d x\right )^{\frac{27}{2}} b^{6} + 237006 \, \left (d x\right )^{\frac{23}{2}} a b^{5} d^{2} + 717255 \, \left (d x\right )^{\frac{19}{2}} a^{2} b^{4} d^{4} + 1211364 \, \left (d x\right )^{\frac{15}{2}} a^{3} b^{3} d^{6} + 1238895 \, \left (d x\right )^{\frac{11}{2}} a^{4} b^{2} d^{8} + 778734 \, \left (d x\right )^{\frac{7}{2}} a^{5} b d^{10} + 302841 \, \left (d x\right )^{\frac{3}{2}} a^{6} d^{12}\right )}}{908523 \, d^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x, algorithm="maxima")

[Out]

2/908523*(33649*(d*x)^(27/2)*b^6 + 237006*(d*x)^(23/2)*a*b^5*d^2 + 717255*(d*x)^(19/2)*a^2*b^4*d^4 + 1211364*(
d*x)^(15/2)*a^3*b^3*d^6 + 1238895*(d*x)^(11/2)*a^4*b^2*d^8 + 778734*(d*x)^(7/2)*a^5*b*d^10 + 302841*(d*x)^(3/2
)*a^6*d^12)/d^13

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Fricas [A]  time = 1.25841, size = 211, normalized size = 1.61 \begin{align*} \frac{2}{908523} \,{\left (33649 \, b^{6} x^{13} + 237006 \, a b^{5} x^{11} + 717255 \, a^{2} b^{4} x^{9} + 1211364 \, a^{3} b^{3} x^{7} + 1238895 \, a^{4} b^{2} x^{5} + 778734 \, a^{5} b x^{3} + 302841 \, a^{6} x\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x, algorithm="fricas")

[Out]

2/908523*(33649*b^6*x^13 + 237006*a*b^5*x^11 + 717255*a^2*b^4*x^9 + 1211364*a^3*b^3*x^7 + 1238895*a^4*b^2*x^5
+ 778734*a^5*b*x^3 + 302841*a^6*x)*sqrt(d*x)

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Sympy [A]  time = 2.85401, size = 131, normalized size = 1. \begin{align*} \frac{2 a^{6} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{12 a^{5} b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{30 a^{4} b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} + \frac{8 a^{3} b^{3} \sqrt{d} x^{\frac{15}{2}}}{3} + \frac{30 a^{2} b^{4} \sqrt{d} x^{\frac{19}{2}}}{19} + \frac{12 a b^{5} \sqrt{d} x^{\frac{23}{2}}}{23} + \frac{2 b^{6} \sqrt{d} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3*(d*x)**(1/2),x)

[Out]

2*a**6*sqrt(d)*x**(3/2)/3 + 12*a**5*b*sqrt(d)*x**(7/2)/7 + 30*a**4*b**2*sqrt(d)*x**(11/2)/11 + 8*a**3*b**3*sqr
t(d)*x**(15/2)/3 + 30*a**2*b**4*sqrt(d)*x**(19/2)/19 + 12*a*b**5*sqrt(d)*x**(23/2)/23 + 2*b**6*sqrt(d)*x**(27/
2)/27

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Giac [A]  time = 1.12483, size = 153, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (33649 \, \sqrt{d x} b^{6} d x^{13} + 237006 \, \sqrt{d x} a b^{5} d x^{11} + 717255 \, \sqrt{d x} a^{2} b^{4} d x^{9} + 1211364 \, \sqrt{d x} a^{3} b^{3} d x^{7} + 1238895 \, \sqrt{d x} a^{4} b^{2} d x^{5} + 778734 \, \sqrt{d x} a^{5} b d x^{3} + 302841 \, \sqrt{d x} a^{6} d x\right )}}{908523 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x, algorithm="giac")

[Out]

2/908523*(33649*sqrt(d*x)*b^6*d*x^13 + 237006*sqrt(d*x)*a*b^5*d*x^11 + 717255*sqrt(d*x)*a^2*b^4*d*x^9 + 121136
4*sqrt(d*x)*a^3*b^3*d*x^7 + 1238895*sqrt(d*x)*a^4*b^2*d*x^5 + 778734*sqrt(d*x)*a^5*b*d*x^3 + 302841*sqrt(d*x)*
a^6*d*x)/d