3.685 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{(d x)^{7/2}} \, dx\)

Optimal. Leaf size=127 \[ \frac{30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac{40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac{10 a^4 b^2 (d x)^{3/2}}{d^5}-\frac{12 a^5 b}{d^3 \sqrt{d x}}-\frac{2 a^6}{5 d (d x)^{5/2}}+\frac{4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac{2 b^6 (d x)^{19/2}}{19 d^{13}} \]

[Out]

(-2*a^6)/(5*d*(d*x)^(5/2)) - (12*a^5*b)/(d^3*Sqrt[d*x]) + (10*a^4*b^2*(d*x)^(3/2))/d^5 + (40*a^3*b^3*(d*x)^(7/
2))/(7*d^7) + (30*a^2*b^4*(d*x)^(11/2))/(11*d^9) + (4*a*b^5*(d*x)^(15/2))/(5*d^11) + (2*b^6*(d*x)^(19/2))/(19*
d^13)

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Rubi [A]  time = 0.0637789, antiderivative size = 127, 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} \[ \frac{30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac{40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac{10 a^4 b^2 (d x)^{3/2}}{d^5}-\frac{12 a^5 b}{d^3 \sqrt{d x}}-\frac{2 a^6}{5 d (d x)^{5/2}}+\frac{4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac{2 b^6 (d x)^{19/2}}{19 d^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0245415, size = 82, normalized size = 0.65 \[ \frac{2 \sqrt{d x} \left (9975 a^2 b^4 x^8+20900 a^3 b^3 x^6+36575 a^4 b^2 x^4-43890 a^5 b x^2-1463 a^6+2926 a b^5 x^{10}+385 b^6 x^{12}\right )}{7315 d^4 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(7/2),x]

[Out]

(2*Sqrt[d*x]*(-1463*a^6 - 43890*a^5*b*x^2 + 36575*a^4*b^2*x^4 + 20900*a^3*b^3*x^6 + 9975*a^2*b^4*x^8 + 2926*a*
b^5*x^10 + 385*b^6*x^12))/(7315*d^4*x^3)

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Maple [A]  time = 0.05, size = 74, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -770\,{b}^{6}{x}^{12}-5852\,a{b}^{5}{x}^{10}-19950\,{a}^{2}{b}^{4}{x}^{8}-41800\,{a}^{3}{b}^{3}{x}^{6}-73150\,{a}^{4}{b}^{2}{x}^{4}+87780\,{a}^{5}b{x}^{2}+2926\,{a}^{6} \right ) x}{7315} \left ( dx \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

-2/7315*(-385*b^6*x^12-2926*a*b^5*x^10-9975*a^2*b^4*x^8-20900*a^3*b^3*x^6-36575*a^4*b^2*x^4+43890*a^5*b*x^2+14
63*a^6)*x/(d*x)^(7/2)

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Maxima [A]  time = 0.991902, size = 154, normalized size = 1.21 \begin{align*} -\frac{2 \,{\left (\frac{1463 \,{\left (30 \, a^{5} b d^{2} x^{2} + a^{6} d^{2}\right )}}{\left (d x\right )^{\frac{5}{2}} d^{2}} - \frac{385 \, \left (d x\right )^{\frac{19}{2}} b^{6} + 2926 \, \left (d x\right )^{\frac{15}{2}} a b^{5} d^{2} + 9975 \, \left (d x\right )^{\frac{11}{2}} a^{2} b^{4} d^{4} + 20900 \, \left (d x\right )^{\frac{7}{2}} a^{3} b^{3} d^{6} + 36575 \, \left (d x\right )^{\frac{3}{2}} a^{4} b^{2} d^{8}}{d^{12}}\right )}}{7315 \, 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)^(7/2),x, algorithm="maxima")

[Out]

-2/7315*(1463*(30*a^5*b*d^2*x^2 + a^6*d^2)/((d*x)^(5/2)*d^2) - (385*(d*x)^(19/2)*b^6 + 2926*(d*x)^(15/2)*a*b^5
*d^2 + 9975*(d*x)^(11/2)*a^2*b^4*d^4 + 20900*(d*x)^(7/2)*a^3*b^3*d^6 + 36575*(d*x)^(3/2)*a^4*b^2*d^8)/d^12)/d

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Fricas [A]  time = 1.19204, size = 201, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (385 \, b^{6} x^{12} + 2926 \, a b^{5} x^{10} + 9975 \, a^{2} b^{4} x^{8} + 20900 \, a^{3} b^{3} x^{6} + 36575 \, a^{4} b^{2} x^{4} - 43890 \, a^{5} b x^{2} - 1463 \, a^{6}\right )} \sqrt{d x}}{7315 \, d^{4} x^{3}} \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)^(7/2),x, algorithm="fricas")

[Out]

2/7315*(385*b^6*x^12 + 2926*a*b^5*x^10 + 9975*a^2*b^4*x^8 + 20900*a^3*b^3*x^6 + 36575*a^4*b^2*x^4 - 43890*a^5*
b*x^2 - 1463*a^6)*sqrt(d*x)/(d^4*x^3)

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Sympy [A]  time = 5.08522, size = 128, normalized size = 1.01 \begin{align*} - \frac{2 a^{6}}{5 d^{\frac{7}{2}} x^{\frac{5}{2}}} - \frac{12 a^{5} b}{d^{\frac{7}{2}} \sqrt{x}} + \frac{10 a^{4} b^{2} x^{\frac{3}{2}}}{d^{\frac{7}{2}}} + \frac{40 a^{3} b^{3} x^{\frac{7}{2}}}{7 d^{\frac{7}{2}}} + \frac{30 a^{2} b^{4} x^{\frac{11}{2}}}{11 d^{\frac{7}{2}}} + \frac{4 a b^{5} x^{\frac{15}{2}}}{5 d^{\frac{7}{2}}} + \frac{2 b^{6} x^{\frac{19}{2}}}{19 d^{\frac{7}{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)**(7/2),x)

[Out]

-2*a**6/(5*d**(7/2)*x**(5/2)) - 12*a**5*b/(d**(7/2)*sqrt(x)) + 10*a**4*b**2*x**(3/2)/d**(7/2) + 40*a**3*b**3*x
**(7/2)/(7*d**(7/2)) + 30*a**2*b**4*x**(11/2)/(11*d**(7/2)) + 4*a*b**5*x**(15/2)/(5*d**(7/2)) + 2*b**6*x**(19/
2)/(19*d**(7/2))

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Giac [A]  time = 1.12686, size = 180, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (\frac{1463 \,{\left (30 \, a^{5} b d^{3} x^{2} + a^{6} d^{3}\right )}}{\sqrt{d x} d^{2} x^{2}} - \frac{385 \, \sqrt{d x} b^{6} d^{171} x^{9} + 2926 \, \sqrt{d x} a b^{5} d^{171} x^{7} + 9975 \, \sqrt{d x} a^{2} b^{4} d^{171} x^{5} + 20900 \, \sqrt{d x} a^{3} b^{3} d^{171} x^{3} + 36575 \, \sqrt{d x} a^{4} b^{2} d^{171} x}{d^{171}}\right )}}{7315 \, d^{4}} \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)^(7/2),x, algorithm="giac")

[Out]

-2/7315*(1463*(30*a^5*b*d^3*x^2 + a^6*d^3)/(sqrt(d*x)*d^2*x^2) - (385*sqrt(d*x)*b^6*d^171*x^9 + 2926*sqrt(d*x)
*a*b^5*d^171*x^7 + 9975*sqrt(d*x)*a^2*b^4*d^171*x^5 + 20900*sqrt(d*x)*a^3*b^3*d^171*x^3 + 36575*sqrt(d*x)*a^4*
b^2*d^171*x)/d^171)/d^4