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

Optimal. Leaf size=129 \[ \frac{30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac{40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac{10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac{12 a^5 b (d x)^{5/2}}{5 d^3}+\frac{2 a^6 \sqrt{d x}}{d}+\frac{4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac{2 b^6 (d x)^{25/2}}{25 d^{13}} \]

[Out]

(2*a^6*Sqrt[d*x])/d + (12*a^5*b*(d*x)^(5/2))/(5*d^3) + (10*a^4*b^2*(d*x)^(9/2))/(3*d^5) + (40*a^3*b^3*(d*x)^(1
3/2))/(13*d^7) + (30*a^2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a*b^5*(d*x)^(21/2))/(7*d^11) + (2*b^6*(d*x)^(25/2))/(
25*d^13)

________________________________________________________________________________________

Rubi [A]  time = 0.0597216, antiderivative size = 129, 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} \[ \frac{30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac{40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac{10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac{12 a^5 b (d x)^{5/2}}{5 d^3}+\frac{2 a^6 \sqrt{d x}}{d}+\frac{4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac{2 b^6 (d x)^{25/2}}{25 d^{13}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^6*Sqrt[d*x])/d + (12*a^5*b*(d*x)^(5/2))/(5*d^3) + (10*a^4*b^2*(d*x)^(9/2))/(3*d^5) + (40*a^3*b^3*(d*x)^(1
3/2))/(13*d^7) + (30*a^2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a*b^5*(d*x)^(21/2))/(7*d^11) + (2*b^6*(d*x)^(25/2))/(
25*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}{\sqrt{d x}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{\sqrt{d x}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{\sqrt{d x}}+\frac{6 a^5 b^7 (d x)^{3/2}}{d^2}+\frac{15 a^4 b^8 (d x)^{7/2}}{d^4}+\frac{20 a^3 b^9 (d x)^{11/2}}{d^6}+\frac{15 a^2 b^{10} (d x)^{15/2}}{d^8}+\frac{6 a b^{11} (d x)^{19/2}}{d^{10}}+\frac{b^{12} (d x)^{23/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac{2 a^6 \sqrt{d x}}{d}+\frac{12 a^5 b (d x)^{5/2}}{5 d^3}+\frac{10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac{40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac{30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac{4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac{2 b^6 (d x)^{25/2}}{25 d^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0216615, size = 77, normalized size = 0.6 \[ \frac{2 \left (102375 a^2 b^4 x^9+178500 a^3 b^3 x^7+193375 a^4 b^2 x^5+139230 a^5 b x^3+116025 a^6 x+33150 a b^5 x^{11}+4641 b^6 x^{13}\right )}{116025 \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(116025*a^6*x + 139230*a^5*b*x^3 + 193375*a^4*b^2*x^5 + 178500*a^3*b^3*x^7 + 102375*a^2*b^4*x^9 + 33150*a*b
^5*x^11 + 4641*b^6*x^13))/(116025*Sqrt[d*x])

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 74, normalized size = 0.6 \begin{align*}{\frac{ \left ( 9282\,{b}^{6}{x}^{12}+66300\,a{b}^{5}{x}^{10}+204750\,{a}^{2}{b}^{4}{x}^{8}+357000\,{a}^{3}{b}^{3}{x}^{6}+386750\,{a}^{4}{b}^{2}{x}^{4}+278460\,{a}^{5}b{x}^{2}+232050\,{a}^{6} \right ) x}{116025}{\frac{1}{\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)^3/(d*x)^(1/2),x)

[Out]

2/116025*(4641*b^6*x^12+33150*a*b^5*x^10+102375*a^2*b^4*x^8+178500*a^3*b^3*x^6+193375*a^4*b^2*x^4+139230*a^5*b
*x^2+116025*a^6)*x/(d*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 0.987749, size = 209, normalized size = 1.62 \begin{align*} \frac{2 \,{\left (116025 \, \sqrt{d x} a^{6} + \frac{4641 \, \left (d x\right )^{\frac{25}{2}} b^{6}}{d^{12}} + \frac{33150 \, \left (d x\right )^{\frac{21}{2}} a b^{5}}{d^{10}} + \frac{81900 \, \left (d x\right )^{\frac{17}{2}} a^{2} b^{4}}{d^{8}} + \frac{71400 \, \left (d x\right )^{\frac{13}{2}} a^{3} b^{3}}{d^{6}} + 7735 \,{\left (\frac{5 \, \left (d x\right )^{\frac{9}{2}} b^{2}}{d^{4}} + \frac{18 \, \left (d x\right )^{\frac{5}{2}} a b}{d^{2}}\right )} a^{4} + 175 \,{\left (\frac{117 \, \left (d x\right )^{\frac{17}{2}} b^{4}}{d^{8}} + \frac{612 \, \left (d x\right )^{\frac{13}{2}} a b^{3}}{d^{6}} + \frac{884 \, \left (d x\right )^{\frac{9}{2}} a^{2} b^{2}}{d^{4}}\right )} a^{2}\right )}}{116025 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/116025*(116025*sqrt(d*x)*a^6 + 4641*(d*x)^(25/2)*b^6/d^12 + 33150*(d*x)^(21/2)*a*b^5/d^10 + 81900*(d*x)^(17/
2)*a^2*b^4/d^8 + 71400*(d*x)^(13/2)*a^3*b^3/d^6 + 7735*(5*(d*x)^(9/2)*b^2/d^4 + 18*(d*x)^(5/2)*a*b/d^2)*a^4 +
175*(117*(d*x)^(17/2)*b^4/d^8 + 612*(d*x)^(13/2)*a*b^3/d^6 + 884*(d*x)^(9/2)*a^2*b^2/d^4)*a^2)/d

________________________________________________________________________________________

Fricas [A]  time = 1.27325, size = 205, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (4641 \, b^{6} x^{12} + 33150 \, a b^{5} x^{10} + 102375 \, a^{2} b^{4} x^{8} + 178500 \, a^{3} b^{3} x^{6} + 193375 \, a^{4} b^{2} x^{4} + 139230 \, a^{5} b x^{2} + 116025 \, a^{6}\right )} \sqrt{d x}}{116025 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/116025*(4641*b^6*x^12 + 33150*a*b^5*x^10 + 102375*a^2*b^4*x^8 + 178500*a^3*b^3*x^6 + 193375*a^4*b^2*x^4 + 13
9230*a^5*b*x^2 + 116025*a^6)*sqrt(d*x)/d

________________________________________________________________________________________

Sympy [A]  time = 2.69479, size = 129, normalized size = 1. \begin{align*} \frac{2 a^{6} \sqrt{x}}{\sqrt{d}} + \frac{12 a^{5} b x^{\frac{5}{2}}}{5 \sqrt{d}} + \frac{10 a^{4} b^{2} x^{\frac{9}{2}}}{3 \sqrt{d}} + \frac{40 a^{3} b^{3} x^{\frac{13}{2}}}{13 \sqrt{d}} + \frac{30 a^{2} b^{4} x^{\frac{17}{2}}}{17 \sqrt{d}} + \frac{4 a b^{5} x^{\frac{21}{2}}}{7 \sqrt{d}} + \frac{2 b^{6} x^{\frac{25}{2}}}{25 \sqrt{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)**(1/2),x)

[Out]

2*a**6*sqrt(x)/sqrt(d) + 12*a**5*b*x**(5/2)/(5*sqrt(d)) + 10*a**4*b**2*x**(9/2)/(3*sqrt(d)) + 40*a**3*b**3*x**
(13/2)/(13*sqrt(d)) + 30*a**2*b**4*x**(17/2)/(17*sqrt(d)) + 4*a*b**5*x**(21/2)/(7*sqrt(d)) + 2*b**6*x**(25/2)/
(25*sqrt(d))

________________________________________________________________________________________

Giac [A]  time = 1.1185, size = 142, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (4641 \, \sqrt{d x} b^{6} x^{12} + 33150 \, \sqrt{d x} a b^{5} x^{10} + 102375 \, \sqrt{d x} a^{2} b^{4} x^{8} + 178500 \, \sqrt{d x} a^{3} b^{3} x^{6} + 193375 \, \sqrt{d x} a^{4} b^{2} x^{4} + 139230 \, \sqrt{d x} a^{5} b x^{2} + 116025 \, \sqrt{d x} a^{6}\right )}}{116025 \, 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)^(1/2),x, algorithm="giac")

[Out]

2/116025*(4641*sqrt(d*x)*b^6*x^12 + 33150*sqrt(d*x)*a*b^5*x^10 + 102375*sqrt(d*x)*a^2*b^4*x^8 + 178500*sqrt(d*
x)*a^3*b^3*x^6 + 193375*sqrt(d*x)*a^4*b^2*x^4 + 139230*sqrt(d*x)*a^5*b*x^2 + 116025*sqrt(d*x)*a^6)/d