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

Optimal. Leaf size=131 \[ \frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]

[Out]

(2*a^6*(d*x)^(5/2))/(5*d) + (4*a^5*b*(d*x)^(9/2))/(3*d^3) + (30*a^4*b^2*(d*x)^(13/2))/(13*d^5) + (40*a^3*b^3*(
d*x)^(17/2))/(17*d^7) + (10*a^2*b^4*(d*x)^(21/2))/(7*d^9) + (12*a*b^5*(d*x)^(25/2))/(25*d^11) + (2*b^6*(d*x)^(
29/2))/(29*d^13)

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Rubi [A]  time = 0.0613499, 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} \[ \frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^6*(d*x)^(5/2))/(5*d) + (4*a^5*b*(d*x)^(9/2))/(3*d^3) + (30*a^4*b^2*(d*x)^(13/2))/(13*d^5) + (40*a^3*b^3*(
d*x)^(17/2))/(17*d^7) + (10*a^2*b^4*(d*x)^(21/2))/(7*d^9) + (12*a*b^5*(d*x)^(25/2))/(25*d^11) + (2*b^6*(d*x)^(
29/2))/(29*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 (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int (d x)^{3/2} \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 (d x)^{3/2}+\frac{6 a^5 b^7 (d x)^{7/2}}{d^2}+\frac{15 a^4 b^8 (d x)^{11/2}}{d^4}+\frac{20 a^3 b^9 (d x)^{15/2}}{d^6}+\frac{15 a^2 b^{10} (d x)^{19/2}}{d^8}+\frac{6 a b^{11} (d x)^{23/2}}{d^{10}}+\frac{b^{12} (d x)^{27/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0247262, size = 77, normalized size = 0.59 \[ \frac{2 x (d x)^{3/2} \left (2403375 a^2 b^4 x^8+3958500 a^3 b^3 x^6+3882375 a^4 b^2 x^4+2243150 a^5 b x^2+672945 a^6+807534 a b^5 x^{10}+116025 b^6 x^{12}\right )}{3364725} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(3/2)*(672945*a^6 + 2243150*a^5*b*x^2 + 3882375*a^4*b^2*x^4 + 3958500*a^3*b^3*x^6 + 2403375*a^2*b^4
*x^8 + 807534*a*b^5*x^10 + 116025*b^6*x^12))/3364725

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Maple [A]  time = 0.05, size = 74, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 116025\,{b}^{6}{x}^{12}+807534\,a{b}^{5}{x}^{10}+2403375\,{a}^{2}{b}^{4}{x}^{8}+3958500\,{a}^{3}{b}^{3}{x}^{6}+3882375\,{a}^{4}{b}^{2}{x}^{4}+2243150\,{a}^{5}b{x}^{2}+672945\,{a}^{6} \right ) }{3364725} \left ( dx \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3364725*x*(116025*b^6*x^12+807534*a*b^5*x^10+2403375*a^2*b^4*x^8+3958500*a^3*b^3*x^6+3882375*a^4*b^2*x^4+224
3150*a^5*b*x^2+672945*a^6)*(d*x)^(3/2)

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Maxima [A]  time = 0.982645, size = 142, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (116025 \, \left (d x\right )^{\frac{29}{2}} b^{6} + 807534 \, \left (d x\right )^{\frac{25}{2}} a b^{5} d^{2} + 2403375 \, \left (d x\right )^{\frac{21}{2}} a^{2} b^{4} d^{4} + 3958500 \, \left (d x\right )^{\frac{17}{2}} a^{3} b^{3} d^{6} + 3882375 \, \left (d x\right )^{\frac{13}{2}} a^{4} b^{2} d^{8} + 2243150 \, \left (d x\right )^{\frac{9}{2}} a^{5} b d^{10} + 672945 \, \left (d x\right )^{\frac{5}{2}} a^{6} d^{12}\right )}}{3364725 \, d^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3364725*(116025*(d*x)^(29/2)*b^6 + 807534*(d*x)^(25/2)*a*b^5*d^2 + 2403375*(d*x)^(21/2)*a^2*b^4*d^4 + 395850
0*(d*x)^(17/2)*a^3*b^3*d^6 + 3882375*(d*x)^(13/2)*a^4*b^2*d^8 + 2243150*(d*x)^(9/2)*a^5*b*d^10 + 672945*(d*x)^
(5/2)*a^6*d^12)/d^13

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Fricas [A]  time = 1.24482, size = 239, normalized size = 1.82 \begin{align*} \frac{2}{3364725} \,{\left (116025 \, b^{6} d x^{14} + 807534 \, a b^{5} d x^{12} + 2403375 \, a^{2} b^{4} d x^{10} + 3958500 \, a^{3} b^{3} d x^{8} + 3882375 \, a^{4} b^{2} d x^{6} + 2243150 \, a^{5} b d x^{4} + 672945 \, a^{6} d x^{2}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3364725*(116025*b^6*d*x^14 + 807534*a*b^5*d*x^12 + 2403375*a^2*b^4*d*x^10 + 3958500*a^3*b^3*d*x^8 + 3882375*
a^4*b^2*d*x^6 + 2243150*a^5*b*d*x^4 + 672945*a^6*d*x^2)*sqrt(d*x)

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Sympy [A]  time = 5.61801, size = 131, normalized size = 1. \begin{align*} \frac{2 a^{6} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a^{5} b d^{\frac{3}{2}} x^{\frac{9}{2}}}{3} + \frac{30 a^{4} b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} + \frac{40 a^{3} b^{3} d^{\frac{3}{2}} x^{\frac{17}{2}}}{17} + \frac{10 a^{2} b^{4} d^{\frac{3}{2}} x^{\frac{21}{2}}}{7} + \frac{12 a b^{5} d^{\frac{3}{2}} x^{\frac{25}{2}}}{25} + \frac{2 b^{6} d^{\frac{3}{2}} x^{\frac{29}{2}}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**6*d**(3/2)*x**(5/2)/5 + 4*a**5*b*d**(3/2)*x**(9/2)/3 + 30*a**4*b**2*d**(3/2)*x**(13/2)/13 + 40*a**3*b**3*
d**(3/2)*x**(17/2)/17 + 10*a**2*b**4*d**(3/2)*x**(21/2)/7 + 12*a*b**5*d**(3/2)*x**(25/2)/25 + 2*b**6*d**(3/2)*
x**(29/2)/29

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Giac [A]  time = 1.15185, size = 149, normalized size = 1.14 \begin{align*} \frac{2}{29} \, \sqrt{d x} b^{6} d x^{14} + \frac{12}{25} \, \sqrt{d x} a b^{5} d x^{12} + \frac{10}{7} \, \sqrt{d x} a^{2} b^{4} d x^{10} + \frac{40}{17} \, \sqrt{d x} a^{3} b^{3} d x^{8} + \frac{30}{13} \, \sqrt{d x} a^{4} b^{2} d x^{6} + \frac{4}{3} \, \sqrt{d x} a^{5} b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{6} d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29*sqrt(d*x)*b^6*d*x^14 + 12/25*sqrt(d*x)*a*b^5*d*x^12 + 10/7*sqrt(d*x)*a^2*b^4*d*x^10 + 40/17*sqrt(d*x)*a^3
*b^3*d*x^8 + 30/13*sqrt(d*x)*a^4*b^2*d*x^6 + 4/3*sqrt(d*x)*a^5*b*d*x^4 + 2/5*sqrt(d*x)*a^6*d*x^2