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

Optimal. Leaf size=293 \[ \frac{2 b^5 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac{4 a^3 b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}+\frac{10 a^4 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]

[Out]

(-2*a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (10*a^4*b*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (4*a^3*b^2*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(a + b*x^2))
+ (20*a^2*b^3*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(13/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(13*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(17/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*d^1
1*(a + b*x^2))

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Rubi [A]  time = 0.0819537, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1112, 270} \[ \frac{2 b^5 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac{4 a^3 b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}+\frac{10 a^4 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (10*a^4*b*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (4*a^3*b^2*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(a + b*x^2))
+ (20*a^2*b^3*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(13/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(13*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(17/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*d^1
1*(a + b*x^2))

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

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

Mathematica [A]  time = 0.0378307, size = 88, normalized size = 0.3 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (2210 a^2 b^3 x^6+3978 a^3 b^2 x^4+9945 a^4 b x^2-663 a^5+765 a b^4 x^8+117 b^5 x^{10}\right )}{1989 (d x)^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[(a + b*x^2)^2]*(-663*a^5 + 9945*a^4*b*x^2 + 3978*a^3*b^2*x^4 + 2210*a^2*b^3*x^6 + 765*a*b^4*x^8 + 11
7*b^5*x^10))/(1989*(d*x)^(5/2)*(a + b*x^2))

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Maple [A]  time = 0.178, size = 83, normalized size = 0.3 \begin{align*} -{\frac{2\, \left ( -117\,{b}^{5}{x}^{10}-765\,a{b}^{4}{x}^{8}-2210\,{a}^{2}{b}^{3}{x}^{6}-3978\,{b}^{2}{a}^{3}{x}^{4}-9945\,{a}^{4}b{x}^{2}+663\,{a}^{5} \right ) x}{1989\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( dx \right ) ^{-{\frac{5}{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)^(5/2)/(d*x)^(5/2),x)

[Out]

-2/1989*x*(-117*b^5*x^10-765*a*b^4*x^8-2210*a^2*b^3*x^6-3978*a^3*b^2*x^4-9945*a^4*b*x^2+663*a^5)*((b*x^2+a)^2)
^(5/2)/(b*x^2+a)^5/(d*x)^(5/2)

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Maxima [A]  time = 1.02199, size = 204, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (45 \,{\left (13 \, b^{5} \sqrt{d} x^{3} + 17 \, a b^{4} \sqrt{d} x\right )} x^{\frac{11}{2}} + 340 \,{\left (9 \, a b^{4} \sqrt{d} x^{3} + 13 \, a^{2} b^{3} \sqrt{d} x\right )} x^{\frac{7}{2}} + 1326 \,{\left (5 \, a^{2} b^{3} \sqrt{d} x^{3} + 9 \, a^{3} b^{2} \sqrt{d} x\right )} x^{\frac{3}{2}} + \frac{7956 \,{\left (a^{3} b^{2} \sqrt{d} x^{3} + 5 \, a^{4} b \sqrt{d} x\right )}}{\sqrt{x}} + \frac{3315 \,{\left (3 \, a^{4} b \sqrt{d} x^{3} - a^{5} \sqrt{d} x\right )}}{x^{\frac{5}{2}}}\right )}}{9945 \, d^{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)^(5/2)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

2/9945*(45*(13*b^5*sqrt(d)*x^3 + 17*a*b^4*sqrt(d)*x)*x^(11/2) + 340*(9*a*b^4*sqrt(d)*x^3 + 13*a^2*b^3*sqrt(d)*
x)*x^(7/2) + 1326*(5*a^2*b^3*sqrt(d)*x^3 + 9*a^3*b^2*sqrt(d)*x)*x^(3/2) + 7956*(a^3*b^2*sqrt(d)*x^3 + 5*a^4*b*
sqrt(d)*x)/sqrt(x) + 3315*(3*a^4*b*sqrt(d)*x^3 - a^5*sqrt(d)*x)/x^(5/2))/d^3

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Fricas [A]  time = 1.47365, size = 167, normalized size = 0.57 \begin{align*} \frac{2 \,{\left (117 \, b^{5} x^{10} + 765 \, a b^{4} x^{8} + 2210 \, a^{2} b^{3} x^{6} + 3978 \, a^{3} b^{2} x^{4} + 9945 \, a^{4} b x^{2} - 663 \, a^{5}\right )} \sqrt{d x}}{1989 \, d^{3} x^{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)^(5/2)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/1989*(117*b^5*x^10 + 765*a*b^4*x^8 + 2210*a^2*b^3*x^6 + 3978*a^3*b^2*x^4 + 9945*a^4*b*x^2 - 663*a^5)*sqrt(d*
x)/(d^3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{\left (d x\right )^{\frac{5}{2}}}\, dx \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)**(5/2)/(d*x)**(5/2),x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/(d*x)**(5/2), x)

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Giac [A]  time = 1.14683, size = 215, normalized size = 0.73 \begin{align*} -\frac{2 \,{\left (\frac{663 \, a^{5} d \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{d x} x} - \frac{117 \, \sqrt{d x} b^{5} d^{136} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 765 \, \sqrt{d x} a b^{4} d^{136} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 2210 \, \sqrt{d x} a^{2} b^{3} d^{136} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 3978 \, \sqrt{d x} a^{3} b^{2} d^{136} x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 9945 \, \sqrt{d x} a^{4} b d^{136} \mathrm{sgn}\left (b x^{2} + a\right )}{d^{136}}\right )}}{1989 \, d^{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)^(5/2)/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/1989*(663*a^5*d*sgn(b*x^2 + a)/(sqrt(d*x)*x) - (117*sqrt(d*x)*b^5*d^136*x^8*sgn(b*x^2 + a) + 765*sqrt(d*x)*
a*b^4*d^136*x^6*sgn(b*x^2 + a) + 2210*sqrt(d*x)*a^2*b^3*d^136*x^4*sgn(b*x^2 + a) + 3978*sqrt(d*x)*a^3*b^2*d^13
6*x^2*sgn(b*x^2 + a) + 9945*sqrt(d*x)*a^4*b*d^136*sgn(b*x^2 + a))/d^136)/d^3