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

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

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Rubi [A]  time = 0.08, antiderivative size = 293, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 )} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.04, size = 88, normalized size = 0.30 \begin {gather*} \frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (-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+117 b^5 x^{10}\right )}{1989 (d x)^{5/2} \left (a+b x^2\right )} \end {gather*}

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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IntegrateAlgebraic [A]  time = 74.82, size = 124, normalized size = 0.42 \begin {gather*} \frac {2 \left (a d^2+b d^2 x^2\right ) \left (-663 a^5 d^{10}+9945 a^4 b d^{10} x^2+3978 a^3 b^2 d^{10} x^4+2210 a^2 b^3 d^{10} x^6+765 a b^4 d^{10} x^8+117 b^5 d^{10} x^{10}\right )}{1989 d^{13} (d x)^{3/2} \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d^2 + b*d^2*x^2)*(-663*a^5*d^10 + 9945*a^4*b*d^10*x^2 + 3978*a^3*b^2*d^10*x^4 + 2210*a^2*b^3*d^10*x^6 +
765*a*b^4*d^10*x^8 + 117*b^5*d^10*x^10))/(1989*d^13*(d*x)^(3/2)*Sqrt[(a*d^2 + b*d^2*x^2)^2/d^4])

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fricas [A]  time = 0.94, size = 67, normalized size = 0.23 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.19, size = 159, normalized size = 0.54 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 83, normalized size = 0.28 \begin {gather*} -\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 ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{1989 \left (b \,x^{2}+a \right )^{5} \left (d x \right )^{\frac {5}{2}}} \end {gather*}

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 = 1.54, size = 151, normalized size = 0.52 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 4.56, size = 116, normalized size = 0.40 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (\frac {10\,a^4\,x^2}{d^2}-\frac {2\,a^5}{3\,b\,d^2}+\frac {2\,b^4\,x^{10}}{17\,d^2}+\frac {4\,a^3\,b\,x^4}{d^2}+\frac {10\,a\,b^3\,x^8}{13\,d^2}+\frac {20\,a^2\,b^2\,x^6}{9\,d^2}\right )}{x^3\,\sqrt {d\,x}+\frac {a\,x\,\sqrt {d\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)*((10*a^4*x^2)/d^2 - (2*a^5)/(3*b*d^2) + (2*b^4*x^10)/(17*d^2) + (4*a^3*b*x^
4)/d^2 + (10*a*b^3*x^8)/(13*d^2) + (20*a^2*b^2*x^6)/(9*d^2)))/(x^3*(d*x)^(1/2) + (a*x*(d*x)^(1/2))/b)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{\left (d x\right )^{\frac {5}{2}}}\, dx \end {gather*}

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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