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

Optimal. Leaf size=297 \[ \frac{2 b^5 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac{4 a^2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

[Out]

(2*a^5*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(7*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^5*(a +
b*x^2)) + (4*a^2*b^3*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(19/2
)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(23/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
/(23*d^11*(a + b*x^2))

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Rubi [A]  time = 0.0812924, antiderivative size = 297, 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} \[ \frac{2 b^5 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac{4 a^2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0307575, size = 88, normalized size = 0.3 \[ \frac{2 \sqrt{d x} \sqrt{\left (a+b x^2\right )^2} \left (67298 a^2 b^3 x^7+91770 a^3 b^2 x^5+72105 a^4 b x^3+33649 a^5 x+26565 a b^4 x^9+4389 b^5 x^{11}\right )}{100947 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(33649*a^5*x + 72105*a^4*b*x^3 + 91770*a^3*b^2*x^5 + 67298*a^2*b^3*x^7 + 2656
5*a*b^4*x^9 + 4389*b^5*x^11))/(100947*(a + b*x^2))

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Maple [A]  time = 0.167, size = 83, normalized size = 0.3 \begin{align*}{\frac{2\,x \left ( 4389\,{b}^{5}{x}^{10}+26565\,a{b}^{4}{x}^{8}+67298\,{a}^{2}{b}^{3}{x}^{6}+91770\,{b}^{2}{a}^{3}{x}^{4}+72105\,{a}^{4}b{x}^{2}+33649\,{a}^{5} \right ) }{100947\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}\sqrt{dx}} \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)^(1/2),x)

[Out]

2/100947*x*(4389*b^5*x^10+26565*a*b^4*x^8+67298*a^2*b^3*x^6+91770*a^3*b^2*x^4+72105*a^4*b*x^2+33649*a^5)*((b*x
^2+a)^2)^(5/2)*(d*x)^(1/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.985182, size = 198, normalized size = 0.67 \begin{align*} \frac{2}{437} \,{\left (19 \, b^{5} \sqrt{d} x^{3} + 23 \, a b^{4} \sqrt{d} x\right )} x^{\frac{17}{2}} + \frac{8}{285} \,{\left (15 \, a b^{4} \sqrt{d} x^{3} + 19 \, a^{2} b^{3} \sqrt{d} x\right )} x^{\frac{13}{2}} + \frac{4}{55} \,{\left (11 \, a^{2} b^{3} \sqrt{d} x^{3} + 15 \, a^{3} b^{2} \sqrt{d} x\right )} x^{\frac{9}{2}} + \frac{8}{77} \,{\left (7 \, a^{3} b^{2} \sqrt{d} x^{3} + 11 \, a^{4} b \sqrt{d} x\right )} x^{\frac{5}{2}} + \frac{2}{21} \,{\left (3 \, a^{4} b \sqrt{d} x^{3} + 7 \, a^{5} \sqrt{d} x\right )} \sqrt{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)^(5/2)*(d*x)^(1/2),x, algorithm="maxima")

[Out]

2/437*(19*b^5*sqrt(d)*x^3 + 23*a*b^4*sqrt(d)*x)*x^(17/2) + 8/285*(15*a*b^4*sqrt(d)*x^3 + 19*a^2*b^3*sqrt(d)*x)
*x^(13/2) + 4/55*(11*a^2*b^3*sqrt(d)*x^3 + 15*a^3*b^2*sqrt(d)*x)*x^(9/2) + 8/77*(7*a^3*b^2*sqrt(d)*x^3 + 11*a^
4*b*sqrt(d)*x)*x^(5/2) + 2/21*(3*a^4*b*sqrt(d)*x^3 + 7*a^5*sqrt(d)*x)*sqrt(x)

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Fricas [A]  time = 1.53238, size = 170, normalized size = 0.57 \begin{align*} \frac{2}{100947} \,{\left (4389 \, b^{5} x^{11} + 26565 \, a b^{4} x^{9} + 67298 \, a^{2} b^{3} x^{7} + 91770 \, a^{3} b^{2} x^{5} + 72105 \, a^{4} b x^{3} + 33649 \, a^{5} x\right )} \sqrt{d 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)^(5/2)*(d*x)^(1/2),x, algorithm="fricas")

[Out]

2/100947*(4389*b^5*x^11 + 26565*a*b^4*x^9 + 67298*a^2*b^3*x^7 + 91770*a^3*b^2*x^5 + 72105*a^4*b*x^3 + 33649*a^
5*x)*sqrt(d*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x} \left (\left (a + b x^{2}\right )^{2}\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)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.23254, size = 194, normalized size = 0.65 \begin{align*} \frac{2 \,{\left (4389 \, \sqrt{d x} b^{5} d x^{11} \mathrm{sgn}\left (b x^{2} + a\right ) + 26565 \, \sqrt{d x} a b^{4} d x^{9} \mathrm{sgn}\left (b x^{2} + a\right ) + 67298 \, \sqrt{d x} a^{2} b^{3} d x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 91770 \, \sqrt{d x} a^{3} b^{2} d x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + 72105 \, \sqrt{d x} a^{4} b d x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 33649 \, \sqrt{d x} a^{5} d x \mathrm{sgn}\left (b x^{2} + a\right )\right )}}{100947 \, 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)^(5/2)*(d*x)^(1/2),x, algorithm="giac")

[Out]

2/100947*(4389*sqrt(d*x)*b^5*d*x^11*sgn(b*x^2 + a) + 26565*sqrt(d*x)*a*b^4*d*x^9*sgn(b*x^2 + a) + 67298*sqrt(d
*x)*a^2*b^3*d*x^7*sgn(b*x^2 + a) + 91770*sqrt(d*x)*a^3*b^2*d*x^5*sgn(b*x^2 + a) + 72105*sqrt(d*x)*a^4*b*d*x^3*
sgn(b*x^2 + a) + 33649*sqrt(d*x)*a^5*d*x*sgn(b*x^2 + a))/d