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

Optimal. Leaf size=249 \[ \frac{5 a^2 x \sqrt{a+b x^2} (8 b c-7 a d)}{128 c^4 \left (c+d x^2\right ) (b c-a d)}+\frac{5 a^3 (8 b c-7 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{3/2}}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-7 a d)}{48 c^2 \left (c+d x^2\right )^3 (b c-a d)}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-7 a d)}{192 c^3 \left (c+d x^2\right )^2 (b c-a d)}-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c \left (c+d x^2\right )^4 (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^(7/2))/(8*c*(b*c - a*d)*(c + d*x^2)^4) + ((8*b*c - 7*a*d)*x*(a + b*x^2)^(5/2))/(48*c^2*(b*c
- a*d)*(c + d*x^2)^3) + (5*a*(8*b*c - 7*a*d)*x*(a + b*x^2)^(3/2))/(192*c^3*(b*c - a*d)*(c + d*x^2)^2) + (5*a^2
*(8*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(128*c^4*(b*c - a*d)*(c + d*x^2)) + (5*a^3*(8*b*c - 7*a*d)*ArcTanh[(Sqrt[b
*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.137348, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {382, 378, 377, 208} \[ \frac{5 a^2 x \sqrt{a+b x^2} (8 b c-7 a d)}{128 c^4 \left (c+d x^2\right ) (b c-a d)}+\frac{5 a^3 (8 b c-7 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{3/2}}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-7 a d)}{48 c^2 \left (c+d x^2\right )^3 (b c-a d)}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-7 a d)}{192 c^3 \left (c+d x^2\right )^2 (b c-a d)}-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c \left (c+d x^2\right )^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*(a + b*x^2)^(7/2))/(8*c*(b*c - a*d)*(c + d*x^2)^4) + ((8*b*c - 7*a*d)*x*(a + b*x^2)^(5/2))/(48*c^2*(b*c
- a*d)*(c + d*x^2)^3) + (5*a*(8*b*c - 7*a*d)*x*(a + b*x^2)^(3/2))/(192*c^3*(b*c - a*d)*(c + d*x^2)^2) + (5*a^2
*(8*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(128*c^4*(b*c - a*d)*(c + d*x^2)) + (5*a^3*(8*b*c - 7*a*d)*ArcTanh[(Sqrt[b
*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(3/2))

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

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

Rule 377

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^5} \, dx &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) \int \frac{\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^4} \, dx}{8 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) x \left (a+b x^2\right )^{5/2}}{48 c^2 (b c-a d) \left (c+d x^2\right )^3}+\frac{(5 a (8 b c-7 a d)) \int \frac{\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx}{48 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) x \left (a+b x^2\right )^{5/2}}{48 c^2 (b c-a d) \left (c+d x^2\right )^3}+\frac{5 a (8 b c-7 a d) x \left (a+b x^2\right )^{3/2}}{192 c^3 (b c-a d) \left (c+d x^2\right )^2}+\frac{\left (5 a^2 (8 b c-7 a d)\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{64 c^3 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) x \left (a+b x^2\right )^{5/2}}{48 c^2 (b c-a d) \left (c+d x^2\right )^3}+\frac{5 a (8 b c-7 a d) x \left (a+b x^2\right )^{3/2}}{192 c^3 (b c-a d) \left (c+d x^2\right )^2}+\frac{5 a^2 (8 b c-7 a d) x \sqrt{a+b x^2}}{128 c^4 (b c-a d) \left (c+d x^2\right )}+\frac{\left (5 a^3 (8 b c-7 a d)\right ) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{128 c^4 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) x \left (a+b x^2\right )^{5/2}}{48 c^2 (b c-a d) \left (c+d x^2\right )^3}+\frac{5 a (8 b c-7 a d) x \left (a+b x^2\right )^{3/2}}{192 c^3 (b c-a d) \left (c+d x^2\right )^2}+\frac{5 a^2 (8 b c-7 a d) x \sqrt{a+b x^2}}{128 c^4 (b c-a d) \left (c+d x^2\right )}+\frac{\left (5 a^3 (8 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 c^4 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^{7/2}}{8 c (b c-a d) \left (c+d x^2\right )^4}+\frac{(8 b c-7 a d) x \left (a+b x^2\right )^{5/2}}{48 c^2 (b c-a d) \left (c+d x^2\right )^3}+\frac{5 a (8 b c-7 a d) x \left (a+b x^2\right )^{3/2}}{192 c^3 (b c-a d) \left (c+d x^2\right )^2}+\frac{5 a^2 (8 b c-7 a d) x \sqrt{a+b x^2}}{128 c^4 (b c-a d) \left (c+d x^2\right )}+\frac{5 a^3 (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.00828, size = 306, normalized size = 1.23 \[ \frac{x \left (\frac{15 a^3 \left (c+d x^2\right )^4 (7 a d-8 b c) \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}}-c \left (2 a^2 b^2 c x^2 \left (173 c^2 d x^2+236 c^3+106 c d^2 x^4+25 d^3 x^6\right )+a^3 b \left (-323 c^2 d^2 x^4-21 c^3 d x^2+264 c^4-335 c d^3 x^6-105 d^4 x^8\right )-a^4 d \left (511 c^2 d x^2+279 c^3+385 c d^2 x^4+105 d^3 x^6\right )+8 a b^3 c^2 x^4 \left (34 c^2+13 c d x^2+3 d^2 x^4\right )+16 b^4 c^3 x^6 \left (4 c+d x^2\right )\right )\right )}{384 c^5 \sqrt{a+b x^2} \left (c+d x^2\right )^4 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-(c*(16*b^4*c^3*x^6*(4*c + d*x^2) + 8*a*b^3*c^2*x^4*(34*c^2 + 13*c*d*x^2 + 3*d^2*x^4) + 2*a^2*b^2*c*x^2*(2
36*c^3 + 173*c^2*d*x^2 + 106*c*d^2*x^4 + 25*d^3*x^6) - a^4*d*(279*c^3 + 511*c^2*d*x^2 + 385*c*d^2*x^4 + 105*d^
3*x^6) + a^3*b*(264*c^4 - 21*c^3*d*x^2 - 323*c^2*d^2*x^4 - 335*c*d^3*x^6 - 105*d^4*x^8))) + (15*a^3*(-8*b*c +
7*a*d)*(c + d*x^2)^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]
))/(384*c^5*(-(b*c) + a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^4)

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Maple [B]  time = 0.048, size = 28625, normalized size = 115. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 8.06962, size = 2612, normalized size = 10.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1536*(15*(8*a^3*b*c^5 - 7*a^4*c^4*d + (8*a^3*b*c*d^4 - 7*a^4*d^5)*x^8 + 4*(8*a^3*b*c^2*d^3 - 7*a^4*c*d^4)*x
^6 + 6*(8*a^3*b*c^3*d^2 - 7*a^4*c^2*d^3)*x^4 + 4*(8*a^3*b*c^4*d - 7*a^4*c^3*d^2)*x^2)*sqrt(b*c^2 - a*c*d)*log(
((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*
x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((16*b^4*c^5*d + 8*a*b^3*c^4*d^2 + 26
*a^2*b^2*c^3*d^3 - 155*a^3*b*c^2*d^4 + 105*a^4*c*d^5)*x^7 + (64*b^4*c^6 + 24*a*b^3*c^5*d + 100*a^2*b^2*c^4*d^2
- 573*a^3*b*c^3*d^3 + 385*a^4*c^2*d^4)*x^5 + (208*a*b^3*c^6 + 50*a^2*b^2*c^5*d - 769*a^3*b*c^4*d^2 + 511*a^4*
c^3*d^3)*x^3 + 3*(88*a^2*b^2*c^6 - 181*a^3*b*c^5*d + 93*a^4*c^4*d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^11 - 2*a*b*c^1
0*d + a^2*c^9*d^2 + (b^2*c^7*d^4 - 2*a*b*c^6*d^5 + a^2*c^5*d^6)*x^8 + 4*(b^2*c^8*d^3 - 2*a*b*c^7*d^4 + a^2*c^6
*d^5)*x^6 + 6*(b^2*c^9*d^2 - 2*a*b*c^8*d^3 + a^2*c^7*d^4)*x^4 + 4*(b^2*c^10*d - 2*a*b*c^9*d^2 + a^2*c^8*d^3)*x
^2), -1/768*(15*(8*a^3*b*c^5 - 7*a^4*c^4*d + (8*a^3*b*c*d^4 - 7*a^4*d^5)*x^8 + 4*(8*a^3*b*c^2*d^3 - 7*a^4*c*d^
4)*x^6 + 6*(8*a^3*b*c^3*d^2 - 7*a^4*c^2*d^3)*x^4 + 4*(8*a^3*b*c^4*d - 7*a^4*c^3*d^2)*x^2)*sqrt(-b*c^2 + a*c*d)
*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2
- a^2*c*d)*x)) - 2*((16*b^4*c^5*d + 8*a*b^3*c^4*d^2 + 26*a^2*b^2*c^3*d^3 - 155*a^3*b*c^2*d^4 + 105*a^4*c*d^5)
*x^7 + (64*b^4*c^6 + 24*a*b^3*c^5*d + 100*a^2*b^2*c^4*d^2 - 573*a^3*b*c^3*d^3 + 385*a^4*c^2*d^4)*x^5 + (208*a*
b^3*c^6 + 50*a^2*b^2*c^5*d - 769*a^3*b*c^4*d^2 + 511*a^4*c^3*d^3)*x^3 + 3*(88*a^2*b^2*c^6 - 181*a^3*b*c^5*d +
93*a^4*c^4*d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^11 - 2*a*b*c^10*d + a^2*c^9*d^2 + (b^2*c^7*d^4 - 2*a*b*c^6*d^5 + a^
2*c^5*d^6)*x^8 + 4*(b^2*c^8*d^3 - 2*a*b*c^7*d^4 + a^2*c^6*d^5)*x^6 + 6*(b^2*c^9*d^2 - 2*a*b*c^8*d^3 + a^2*c^7*
d^4)*x^4 + 4*(b^2*c^10*d - 2*a*b*c^9*d^2 + a^2*c^8*d^3)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 4.8351, size = 1955, normalized size = 7.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/128*(8*a^3*b^(3/2)*c - 7*a^4*sqrt(b)*d)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-
b^2*c^2 + a*b*c*d))/((b*c^5 - a*c^4*d)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/192*(120*(sqrt(b)*x - sqrt(b*x^2 + a))^14
*a^3*b^(3/2)*c*d^6 - 105*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^4*sqrt(b)*d^7 - 768*(sqrt(b)*x - sqrt(b*x^2 + a))^
12*b^(11/2)*c^5*d^2 + 768*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(9/2)*c^4*d^3 + 1680*(sqrt(b)*x - sqrt(b*x^2 +
a))^12*a^3*b^(5/2)*c^2*d^5 - 2310*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^4*b^(3/2)*c*d^6 + 735*(sqrt(b)*x - sqrt(b
*x^2 + a))^12*a^5*sqrt(b)*d^7 - 2048*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(13/2)*c^6*d + 2048*(sqrt(b)*x - sqrt(
b*x^2 + a))^10*a^2*b^(9/2)*c^4*d^3 + 8320*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(7/2)*c^3*d^4 - 15600*(sqrt(b
)*x - sqrt(b*x^2 + a))^10*a^4*b^(5/2)*c^2*d^5 + 9800*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^5*b^(3/2)*c*d^6 - 2205
*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^6*sqrt(b)*d^7 - 2048*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(15/2)*c^7 + 1024*(
sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(13/2)*c^6*d - 4864*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(11/2)*c^5*d^2 +
21888*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(9/2)*c^4*d^3 - 38000*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(7/2)*
c^3*d^4 + 37400*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(5/2)*c^2*d^5 - 18550*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^
6*b^(3/2)*c*d^6 + 3675*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^7*sqrt(b)*d^7 - 2048*(sqrt(b)*x - sqrt(b*x^2 + a))^6*
a^2*b^(13/2)*c^6*d - 9472*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(9/2)*c^4*d^3 + 32896*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*a^5*b^(7/2)*c^3*d^4 - 35376*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(5/2)*c^2*d^5 + 18200*(sqrt(b)*x - s
qrt(b*x^2 + a))^6*a^7*b^(3/2)*c*d^6 - 3675*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^8*sqrt(b)*d^7 - 768*(sqrt(b)*x -
sqrt(b*x^2 + a))^4*a^4*b^(11/2)*c^5*d^2 - 1536*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(9/2)*c^4*d^3 - 2944*(sqr
t(b)*x - sqrt(b*x^2 + a))^4*a^6*b^(7/2)*c^3*d^4 + 12528*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(5/2)*c^2*d^5 -
9170*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^8*b^(3/2)*c*d^6 + 2205*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^9*sqrt(b)*d^7
- 256*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*b^(9/2)*c^4*d^3 - 256*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(7/2)*c^
3*d^4 - 608*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^8*b^(5/2)*c^2*d^5 + 1960*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^9*b^(
3/2)*c*d^6 - 735*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^10*sqrt(b)*d^7 - 16*a^8*b^(7/2)*c^3*d^4 - 24*a^9*b^(5/2)*c^
2*d^5 - 50*a^10*b^(3/2)*c*d^6 + 105*a^11*sqrt(b)*d^7)/((b*c^5*d^3 - a*c^4*d^4)*((sqrt(b)*x - sqrt(b*x^2 + a))^
4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^4)