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

Optimal. Leaf size=125 \[ \frac{2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac{40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac{30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac{4 a^5 b (d x)^{3/2}}{d^3}-\frac{2 a^6}{d \sqrt{d x}}+\frac{12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac{2 b^6 (d x)^{23/2}}{23 d^{13}} \]

[Out]

(-2*a^6)/(d*Sqrt[d*x]) + (4*a^5*b*(d*x)^(3/2))/d^3 + (30*a^4*b^2*(d*x)^(7/2))/(7*d^5) + (40*a^3*b^3*(d*x)^(11/
2))/(11*d^7) + (2*a^2*b^4*(d*x)^(15/2))/d^9 + (12*a*b^5*(d*x)^(19/2))/(19*d^11) + (2*b^6*(d*x)^(23/2))/(23*d^1
3)

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Rubi [A]  time = 0.0598052, antiderivative size = 125, 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} \[ \frac{2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac{40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac{30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac{4 a^5 b (d x)^{3/2}}{d^3}-\frac{2 a^6}{d \sqrt{d x}}+\frac{12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac{2 b^6 (d x)^{23/2}}{23 d^{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^6)/(d*Sqrt[d*x]) + (4*a^5*b*(d*x)^(3/2))/d^3 + (30*a^4*b^2*(d*x)^(7/2))/(7*d^5) + (40*a^3*b^3*(d*x)^(11/
2))/(11*d^7) + (2*a^2*b^4*(d*x)^(15/2))/d^9 + (12*a*b^5*(d*x)^(19/2))/(19*d^11) + (2*b^6*(d*x)^(23/2))/(23*d^1
3)

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

Mathematica [A]  time = 0.0216665, size = 77, normalized size = 0.62 \[ \frac{2 x \left (33649 a^2 b^4 x^8+61180 a^3 b^3 x^6+72105 a^4 b^2 x^4+67298 a^5 b x^2-33649 a^6+10626 a b^5 x^{10}+1463 b^6 x^{12}\right )}{33649 (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-33649*a^6 + 67298*a^5*b*x^2 + 72105*a^4*b^2*x^4 + 61180*a^3*b^3*x^6 + 33649*a^2*b^4*x^8 + 10626*a*b^5*x
^10 + 1463*b^6*x^12))/(33649*(d*x)^(3/2))

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Maple [A]  time = 0.048, size = 74, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2926\,{b}^{6}{x}^{12}-21252\,a{b}^{5}{x}^{10}-67298\,{a}^{2}{b}^{4}{x}^{8}-122360\,{a}^{3}{b}^{3}{x}^{6}-144210\,{a}^{4}{b}^{2}{x}^{4}-134596\,{a}^{5}b{x}^{2}+67298\,{a}^{6} \right ) x}{33649} \left ( dx \right ) ^{-{\frac{3}{2}}}} \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)^(3/2),x)

[Out]

-2/33649*(-1463*b^6*x^12-10626*a*b^5*x^10-33649*a^2*b^4*x^8-61180*a^3*b^3*x^6-72105*a^4*b^2*x^4-67298*a^5*b*x^
2+33649*a^6)*x/(d*x)^(3/2)

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Maxima [A]  time = 0.974442, size = 146, normalized size = 1.17 \begin{align*} -\frac{2 \,{\left (\frac{33649 \, a^{6}}{\sqrt{d x}} - \frac{1463 \, \left (d x\right )^{\frac{23}{2}} b^{6} + 10626 \, \left (d x\right )^{\frac{19}{2}} a b^{5} d^{2} + 33649 \, \left (d x\right )^{\frac{15}{2}} a^{2} b^{4} d^{4} + 61180 \, \left (d x\right )^{\frac{11}{2}} a^{3} b^{3} d^{6} + 72105 \, \left (d x\right )^{\frac{7}{2}} a^{4} b^{2} d^{8} + 67298 \, \left (d x\right )^{\frac{3}{2}} a^{5} b d^{10}}{d^{12}}\right )}}{33649 \, 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)^(3/2),x, algorithm="maxima")

[Out]

-2/33649*(33649*a^6/sqrt(d*x) - (1463*(d*x)^(23/2)*b^6 + 10626*(d*x)^(19/2)*a*b^5*d^2 + 33649*(d*x)^(15/2)*a^2
*b^4*d^4 + 61180*(d*x)^(11/2)*a^3*b^3*d^6 + 72105*(d*x)^(7/2)*a^4*b^2*d^8 + 67298*(d*x)^(3/2)*a^5*b*d^10)/d^12
)/d

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Fricas [A]  time = 1.20081, size = 205, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (1463 \, b^{6} x^{12} + 10626 \, a b^{5} x^{10} + 33649 \, a^{2} b^{4} x^{8} + 61180 \, a^{3} b^{3} x^{6} + 72105 \, a^{4} b^{2} x^{4} + 67298 \, a^{5} b x^{2} - 33649 \, a^{6}\right )} \sqrt{d x}}{33649 \, d^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

2/33649*(1463*b^6*x^12 + 10626*a*b^5*x^10 + 33649*a^2*b^4*x^8 + 61180*a^3*b^3*x^6 + 72105*a^4*b^2*x^4 + 67298*
a^5*b*x^2 - 33649*a^6)*sqrt(d*x)/(d^2*x)

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Sympy [A]  time = 2.776, size = 126, normalized size = 1.01 \begin{align*} - \frac{2 a^{6}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{4 a^{5} b x^{\frac{3}{2}}}{d^{\frac{3}{2}}} + \frac{30 a^{4} b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{2}}} + \frac{40 a^{3} b^{3} x^{\frac{11}{2}}}{11 d^{\frac{3}{2}}} + \frac{2 a^{2} b^{4} x^{\frac{15}{2}}}{d^{\frac{3}{2}}} + \frac{12 a b^{5} x^{\frac{19}{2}}}{19 d^{\frac{3}{2}}} + \frac{2 b^{6} x^{\frac{23}{2}}}{23 d^{\frac{3}{2}}} \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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.15905, size = 171, normalized size = 1.37 \begin{align*} -\frac{2 \,{\left (\frac{33649 \, a^{6}}{\sqrt{d x}} - \frac{1463 \, \sqrt{d x} b^{6} d^{275} x^{11} + 10626 \, \sqrt{d x} a b^{5} d^{275} x^{9} + 33649 \, \sqrt{d x} a^{2} b^{4} d^{275} x^{7} + 61180 \, \sqrt{d x} a^{3} b^{3} d^{275} x^{5} + 72105 \, \sqrt{d x} a^{4} b^{2} d^{275} x^{3} + 67298 \, \sqrt{d x} a^{5} b d^{275} x}{d^{276}}\right )}}{33649 \, 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)^(3/2),x, algorithm="giac")

[Out]

-2/33649*(33649*a^6/sqrt(d*x) - (1463*sqrt(d*x)*b^6*d^275*x^11 + 10626*sqrt(d*x)*a*b^5*d^275*x^9 + 33649*sqrt(
d*x)*a^2*b^4*d^275*x^7 + 61180*sqrt(d*x)*a^3*b^3*d^275*x^5 + 72105*sqrt(d*x)*a^4*b^2*d^275*x^3 + 67298*sqrt(d*
x)*a^5*b*d^275*x)/d^276)/d