3.679 \(\int (d x)^{5/2} (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)

Optimal. Leaf size=129 \[ \frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}} \]

[Out]

(2*a^6*(d*x)^(7/2))/(7*d) + (12*a^5*b*(d*x)^(11/2))/(11*d^3) + (2*a^4*b^2*(d*x)^(15/2))/d^5 + (40*a^3*b^3*(d*x
)^(19/2))/(19*d^7) + (30*a^2*b^4*(d*x)^(23/2))/(23*d^9) + (4*a*b^5*(d*x)^(27/2))/(9*d^11) + (2*b^6*(d*x)^(31/2
))/(31*d^13)

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Rubi [A]  time = 0.066312, antiderivative size = 129, 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} \[ \frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*a^6*(d*x)^(7/2))/(7*d) + (12*a^5*b*(d*x)^(11/2))/(11*d^3) + (2*a^4*b^2*(d*x)^(15/2))/d^5 + (40*a^3*b^3*(d*x
)^(19/2))/(19*d^7) + (30*a^2*b^4*(d*x)^(23/2))/(23*d^9) + (4*a*b^5*(d*x)^(27/2))/(9*d^11) + (2*b^6*(d*x)^(31/2
))/(31*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)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int (d x)^{5/2} \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 (d x)^{5/2}+\frac{6 a^5 b^7 (d x)^{9/2}}{d^2}+\frac{15 a^4 b^8 (d x)^{13/2}}{d^4}+\frac{20 a^3 b^9 (d x)^{17/2}}{d^6}+\frac{15 a^2 b^{10} (d x)^{21/2}}{d^8}+\frac{6 a b^{11} (d x)^{25/2}}{d^{10}}+\frac{b^{12} (d x)^{29/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0286528, size = 77, normalized size = 0.6 \[ \frac{2 x (d x)^{5/2} \left (6122655 a^2 b^4 x^8+9882180 a^3 b^3 x^6+9388071 a^4 b^2 x^4+5120766 a^5 b x^2+1341153 a^6+2086238 a b^5 x^{10}+302841 b^6 x^{12}\right )}{9388071} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*x*(d*x)^(5/2)*(1341153*a^6 + 5120766*a^5*b*x^2 + 9388071*a^4*b^2*x^4 + 9882180*a^3*b^3*x^6 + 6122655*a^2*b^
4*x^8 + 2086238*a*b^5*x^10 + 302841*b^6*x^12))/9388071

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Maple [A]  time = 0.049, size = 74, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 302841\,{b}^{6}{x}^{12}+2086238\,a{b}^{5}{x}^{10}+6122655\,{a}^{2}{b}^{4}{x}^{8}+9882180\,{a}^{3}{b}^{3}{x}^{6}+9388071\,{a}^{4}{b}^{2}{x}^{4}+5120766\,{a}^{5}b{x}^{2}+1341153\,{a}^{6} \right ) }{9388071} \left ( dx \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/9388071*x*(302841*b^6*x^12+2086238*a*b^5*x^10+6122655*a^2*b^4*x^8+9882180*a^3*b^3*x^6+9388071*a^4*b^2*x^4+51
20766*a^5*b*x^2+1341153*a^6)*(d*x)^(5/2)

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Maxima [A]  time = 0.95892, size = 142, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (302841 \, \left (d x\right )^{\frac{31}{2}} b^{6} + 2086238 \, \left (d x\right )^{\frac{27}{2}} a b^{5} d^{2} + 6122655 \, \left (d x\right )^{\frac{23}{2}} a^{2} b^{4} d^{4} + 9882180 \, \left (d x\right )^{\frac{19}{2}} a^{3} b^{3} d^{6} + 9388071 \, \left (d x\right )^{\frac{15}{2}} a^{4} b^{2} d^{8} + 5120766 \, \left (d x\right )^{\frac{11}{2}} a^{5} b d^{10} + 1341153 \, \left (d x\right )^{\frac{7}{2}} a^{6} d^{12}\right )}}{9388071 \, d^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

2/9388071*(302841*(d*x)^(31/2)*b^6 + 2086238*(d*x)^(27/2)*a*b^5*d^2 + 6122655*(d*x)^(23/2)*a^2*b^4*d^4 + 98821
80*(d*x)^(19/2)*a^3*b^3*d^6 + 9388071*(d*x)^(15/2)*a^4*b^2*d^8 + 5120766*(d*x)^(11/2)*a^5*b*d^10 + 1341153*(d*
x)^(7/2)*a^6*d^12)/d^13

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Fricas [A]  time = 1.22685, size = 261, normalized size = 2.02 \begin{align*} \frac{2}{9388071} \,{\left (302841 \, b^{6} d^{2} x^{15} + 2086238 \, a b^{5} d^{2} x^{13} + 6122655 \, a^{2} b^{4} d^{2} x^{11} + 9882180 \, a^{3} b^{3} d^{2} x^{9} + 9388071 \, a^{4} b^{2} d^{2} x^{7} + 5120766 \, a^{5} b d^{2} x^{5} + 1341153 \, a^{6} d^{2} x^{3}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

2/9388071*(302841*b^6*d^2*x^15 + 2086238*a*b^5*d^2*x^13 + 6122655*a^2*b^4*d^2*x^11 + 9882180*a^3*b^3*d^2*x^9 +
9388071*a^4*b^2*d^2*x^7 + 5120766*a^5*b*d^2*x^5 + 1341153*a^6*d^2*x^3)*sqrt(d*x)

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Sympy [A]  time = 12.5737, size = 129, normalized size = 1. \begin{align*} \frac{2 a^{6} d^{\frac{5}{2}} x^{\frac{7}{2}}}{7} + \frac{12 a^{5} b d^{\frac{5}{2}} x^{\frac{11}{2}}}{11} + 2 a^{4} b^{2} d^{\frac{5}{2}} x^{\frac{15}{2}} + \frac{40 a^{3} b^{3} d^{\frac{5}{2}} x^{\frac{19}{2}}}{19} + \frac{30 a^{2} b^{4} d^{\frac{5}{2}} x^{\frac{23}{2}}}{23} + \frac{4 a b^{5} d^{\frac{5}{2}} x^{\frac{27}{2}}}{9} + \frac{2 b^{6} d^{\frac{5}{2}} x^{\frac{31}{2}}}{31} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

2*a**6*d**(5/2)*x**(7/2)/7 + 12*a**5*b*d**(5/2)*x**(11/2)/11 + 2*a**4*b**2*d**(5/2)*x**(15/2) + 40*a**3*b**3*d
**(5/2)*x**(19/2)/19 + 30*a**2*b**4*d**(5/2)*x**(23/2)/23 + 4*a*b**5*d**(5/2)*x**(27/2)/9 + 2*b**6*d**(5/2)*x*
*(31/2)/31

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Giac [A]  time = 1.1135, size = 167, normalized size = 1.29 \begin{align*} \frac{2}{31} \, \sqrt{d x} b^{6} d^{2} x^{15} + \frac{4}{9} \, \sqrt{d x} a b^{5} d^{2} x^{13} + \frac{30}{23} \, \sqrt{d x} a^{2} b^{4} d^{2} x^{11} + \frac{40}{19} \, \sqrt{d x} a^{3} b^{3} d^{2} x^{9} + 2 \, \sqrt{d x} a^{4} b^{2} d^{2} x^{7} + \frac{12}{11} \, \sqrt{d x} a^{5} b d^{2} x^{5} + \frac{2}{7} \, \sqrt{d x} a^{6} d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

2/31*sqrt(d*x)*b^6*d^2*x^15 + 4/9*sqrt(d*x)*a*b^5*d^2*x^13 + 30/23*sqrt(d*x)*a^2*b^4*d^2*x^11 + 40/19*sqrt(d*x
)*a^3*b^3*d^2*x^9 + 2*sqrt(d*x)*a^4*b^2*d^2*x^7 + 12/11*sqrt(d*x)*a^5*b*d^2*x^5 + 2/7*sqrt(d*x)*a^6*d^2*x^3