3.64 \(\int (a+b x^2)^{5/2} (c+d x^2) \, dx\)

Optimal. Leaf size=149 \[ \frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-a d)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b} \]

[Out]

(5*a^2*(8*b*c - a*d)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^2)^(3/2))/(192*b) + ((8*b*c -
a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sq
rt[a + b*x^2]])/(128*b^(3/2))

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Rubi [A]  time = 0.0522127, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {388, 195, 217, 206} \[ \frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-a d)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)*(c + d*x^2),x]

[Out]

(5*a^2*(8*b*c - a*d)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^2)^(3/2))/(192*b) + ((8*b*c -
a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sq
rt[a + b*x^2]])/(128*b^(3/2))

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx &=\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac{(-8 b c+a d) \int \left (a+b x^2\right )^{5/2} \, dx}{8 b}\\ &=\frac{(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac{(5 a (8 b c-a d)) \int \left (a+b x^2\right )^{3/2} \, dx}{48 b}\\ &=\frac{5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac{\left (5 a^2 (8 b c-a d)\right ) \int \sqrt{a+b x^2} \, dx}{64 b}\\ &=\frac{5 a^2 (8 b c-a d) x \sqrt{a+b x^2}}{128 b}+\frac{5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac{\left (5 a^3 (8 b c-a d)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{128 b}\\ &=\frac{5 a^2 (8 b c-a d) x \sqrt{a+b x^2}}{128 b}+\frac{5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac{\left (5 a^3 (8 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b}\\ &=\frac{5 a^2 (8 b c-a d) x \sqrt{a+b x^2}}{128 b}+\frac{5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.221856, size = 130, normalized size = 0.87 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (2 a^2 b \left (132 c+59 d x^2\right )+15 a^3 d+8 a b^2 x^2 \left (26 c+17 d x^2\right )+16 b^3 x^4 \left (4 c+3 d x^2\right )\right )-\frac{15 a^{5/2} (a d-8 b c) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{384 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^3*d + 16*b^3*x^4*(4*c + 3*d*x^2) + 8*a*b^2*x^2*(26*c + 17*d*x^2) + 2*a^2*b*(
132*c + 59*d*x^2)) - (15*a^(5/2)*(-8*b*c + a*d)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(3/
2))

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Maple [A]  time = 0.003, size = 166, normalized size = 1.1 \begin{align*}{\frac{dx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{adx}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,d{a}^{2}x}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,d{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,d{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{cx}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,acx}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}cx}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,c{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d*x*(b*x^2+a)^(7/2)/b-1/48*d/b*a*x*(b*x^2+a)^(5/2)-5/192*d/b*a^2*x*(b*x^2+a)^(3/2)-5/128*d/b*a^3*x*(b*x^2+
a)^(1/2)-5/128*d/b^(3/2)*a^4*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/6*c*x*(b*x^2+a)^(5/2)+5/24*c*a*x*(b*x^2+a)^(3/2)+
5/16*c*a^2*x*(b*x^2+a)^(1/2)+5/16*c*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.24254, size = 605, normalized size = 4.06 \begin{align*} \left [-\frac{15 \,{\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (48 \, b^{4} d x^{7} + 8 \,{\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \,{\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \,{\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt{b x^{2} + a}}{768 \, b^{2}}, -\frac{15 \,{\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (48 \, b^{4} d x^{7} + 8 \,{\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \,{\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \,{\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt{b x^{2} + a}}{384 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c),x, algorithm="fricas")

[Out]

[-1/768*(15*(8*a^3*b*c - a^4*d)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(48*b^4*d*x^7 + 8*
(8*b^4*c + 17*a*b^3*d)*x^5 + 2*(104*a*b^3*c + 59*a^2*b^2*d)*x^3 + 3*(88*a^2*b^2*c + 5*a^3*b*d)*x)*sqrt(b*x^2 +
a))/b^2, -1/384*(15*(8*a^3*b*c - a^4*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (48*b^4*d*x^7 + 8*(8*b^
4*c + 17*a*b^3*d)*x^5 + 2*(104*a*b^3*c + 59*a^2*b^2*d)*x^3 + 3*(88*a^2*b^2*c + 5*a^3*b*d)*x)*sqrt(b*x^2 + a))/
b^2]

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Sympy [B]  time = 26.2698, size = 316, normalized size = 2.12 \begin{align*} \frac{5 a^{\frac{7}{2}} d x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{5}{2}} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} c x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} d x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} b c x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} b d x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 \sqrt{a} b^{2} c x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{2} d x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 a^{4} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{5 a^{3} c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{b^{3} c x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} d x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**(7/2)*d*x/(128*b*sqrt(1 + b*x**2/a)) + a**(5/2)*c*x*sqrt(1 + b*x**2/a)/2 + 3*a**(5/2)*c*x/(16*sqrt(1 + b*
x**2/a)) + 133*a**(5/2)*d*x**3/(384*sqrt(1 + b*x**2/a)) + 35*a**(3/2)*b*c*x**3/(48*sqrt(1 + b*x**2/a)) + 127*a
**(3/2)*b*d*x**5/(192*sqrt(1 + b*x**2/a)) + 17*sqrt(a)*b**2*c*x**5/(24*sqrt(1 + b*x**2/a)) + 23*sqrt(a)*b**2*d
*x**7/(48*sqrt(1 + b*x**2/a)) - 5*a**4*d*asinh(sqrt(b)*x/sqrt(a))/(128*b**(3/2)) + 5*a**3*c*asinh(sqrt(b)*x/sq
rt(a))/(16*sqrt(b)) + b**3*c*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + b**3*d*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.15324, size = 182, normalized size = 1.22 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} d x^{2} + \frac{8 \, b^{8} c + 17 \, a b^{7} d}{b^{6}}\right )} x^{2} + \frac{104 \, a b^{7} c + 59 \, a^{2} b^{6} d}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (88 \, a^{2} b^{6} c + 5 \, a^{3} b^{5} d\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x - \frac{5 \,{\left (8 \, a^{3} b c - a^{4} d\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*b^2*d*x^2 + (8*b^8*c + 17*a*b^7*d)/b^6)*x^2 + (104*a*b^7*c + 59*a^2*b^6*d)/b^6)*x^2 + 3*(88*a^2
*b^6*c + 5*a^3*b^5*d)/b^6)*sqrt(b*x^2 + a)*x - 5/128*(8*a^3*b*c - a^4*d)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))
)/b^(3/2)