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

Optimal. Leaf size=293 \[ \frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )} \]

[Out]

2*a^4*b*(d*x)^(5/2)*((b*x^2+a)^2)^(1/2)/d^3/(b*x^2+a)+20/9*a^3*b^2*(d*x)^(9/2)*((b*x^2+a)^2)^(1/2)/d^5/(b*x^2+
a)+20/13*a^2*b^3*(d*x)^(13/2)*((b*x^2+a)^2)^(1/2)/d^7/(b*x^2+a)+10/17*a*b^4*(d*x)^(17/2)*((b*x^2+a)^2)^(1/2)/d
^9/(b*x^2+a)+2/21*b^5*(d*x)^(21/2)*((b*x^2+a)^2)^(1/2)/d^11/(b*x^2+a)+2*a^5*(d*x)^(1/2)*((b*x^2+a)^2)^(1/2)/d/
(b*x^2+a)

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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} \[ \frac {2 b^5 (d x)^{21/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac {2 a^4 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {2 a^5 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.03, size = 88, normalized size = 0.30 \[ \frac {2 \sqrt {\left (a+b x^2\right )^2} \left (13923 a^5 x+13923 a^4 b x^3+15470 a^3 b^2 x^5+10710 a^2 b^3 x^7+4095 a b^4 x^9+663 b^5 x^{11}\right )}{13923 \sqrt {d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x^2)^2]*(13923*a^5*x + 13923*a^4*b*x^3 + 15470*a^3*b^2*x^5 + 10710*a^2*b^3*x^7 + 4095*a*b^4*x^9
+ 663*b^5*x^11))/(13923*Sqrt[d*x]*(a + b*x^2))

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fricas [A]  time = 0.91, size = 64, normalized size = 0.22 \[ \frac {2 \, {\left (663 \, b^{5} x^{10} + 4095 \, a b^{4} x^{8} + 10710 \, a^{2} b^{3} x^{6} + 15470 \, a^{3} b^{2} x^{4} + 13923 \, a^{4} b x^{2} + 13923 \, a^{5}\right )} \sqrt {d x}}{13923 \, d} \]

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]

2/13923*(663*b^5*x^10 + 4095*a*b^4*x^8 + 10710*a^2*b^3*x^6 + 15470*a^3*b^2*x^4 + 13923*a^4*b*x^2 + 13923*a^5)*
sqrt(d*x)/d

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giac [A]  time = 0.17, size = 137, normalized size = 0.47 \[ \frac {2 \, {\left (663 \, \sqrt {d x} b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 4095 \, \sqrt {d x} a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 10710 \, \sqrt {d x} a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 15470 \, \sqrt {d x} a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt {d x} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{13923 \, d} \]

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]

2/13923*(663*sqrt(d*x)*b^5*x^10*sgn(b*x^2 + a) + 4095*sqrt(d*x)*a*b^4*x^8*sgn(b*x^2 + a) + 10710*sqrt(d*x)*a^2
*b^3*x^6*sgn(b*x^2 + a) + 15470*sqrt(d*x)*a^3*b^2*x^4*sgn(b*x^2 + a) + 13923*sqrt(d*x)*a^4*b*x^2*sgn(b*x^2 + a
) + 13923*sqrt(d*x)*a^5*sgn(b*x^2 + a))/d

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maple [A]  time = 0.01, size = 83, normalized size = 0.28 \[ \frac {2 \left (663 b^{5} x^{10}+4095 a \,b^{4} x^{8}+10710 a^{2} b^{3} x^{6}+15470 a^{3} b^{2} x^{4}+13923 a^{4} b \,x^{2}+13923 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{13923 \left (b \,x^{2}+a \right )^{5} \sqrt {d x}} \]

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]

2/13923*x*(663*b^5*x^10+4095*a*b^4*x^8+10710*a^2*b^3*x^6+15470*a^3*b^2*x^4+13923*a^4*b*x^2+13923*a^5)*((b*x^2+
a)^2)^(5/2)/(b*x^2+a)^5/(d*x)^(1/2)

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maxima [A]  time = 1.45, size = 151, normalized size = 0.52 \[ \frac {2 \, {\left (195 \, {\left (17 \, b^{5} \sqrt {d} x^{3} + 21 \, a b^{4} \sqrt {d} x\right )} x^{\frac {15}{2}} + 1260 \, {\left (13 \, a b^{4} \sqrt {d} x^{3} + 17 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {11}{2}} + 3570 \, {\left (9 \, a^{2} b^{3} \sqrt {d} x^{3} + 13 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {7}{2}} + 6188 \, {\left (5 \, a^{3} b^{2} \sqrt {d} x^{3} + 9 \, a^{4} b \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {13923 \, {\left (a^{4} b \sqrt {d} x^{3} + 5 \, a^{5} \sqrt {d} x\right )}}{\sqrt {x}}\right )}}{69615 \, d} \]

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]

2/69615*(195*(17*b^5*sqrt(d)*x^3 + 21*a*b^4*sqrt(d)*x)*x^(15/2) + 1260*(13*a*b^4*sqrt(d)*x^3 + 17*a^2*b^3*sqrt
(d)*x)*x^(11/2) + 3570*(9*a^2*b^3*sqrt(d)*x^3 + 13*a^3*b^2*sqrt(d)*x)*x^(7/2) + 6188*(5*a^3*b^2*sqrt(d)*x^3 +
9*a^4*b*sqrt(d)*x)*x^(3/2) + 13923*(a^4*b*sqrt(d)*x^3 + 5*a^5*sqrt(d)*x)/sqrt(x))/d

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mupad [B]  time = 4.57, size = 112, normalized size = 0.38 \[ \frac {2\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (5731\,a^4+8192\,a^3\,b\,x^2+7278\,a^2\,b^2\,x^4+3432\,a\,b^3\,x^6+663\,b^4\,x^8\right )}{13923\,\sqrt {d\,x}}+\frac {16384\,a^5\,x\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{13923\,\sqrt {d\,x}\,\left (b\,x^2+a\right )} \]

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)^(1/2),x)

[Out]

(2*x*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)*(5731*a^4 + 663*b^4*x^8 + 8192*a^3*b*x^2 + 3432*a*b^3*x^6 + 7278*a^2*b^
2*x^4))/(13923*(d*x)^(1/2)) + (16384*a^5*x*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(13923*(d*x)^(1/2)*(a + b*x^2))

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

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

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