3.1.61 \(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^5} \, dx\)

Optimal. Leaf size=300 \[ \frac {a^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{5/2}}+\frac {x \sqrt {a+b x^2} \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{192 c^3 d \left (c+d x^2\right )^2 (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{384 c^4 d \left (c+d x^2\right ) (b c-a d)^2}+\frac {x \sqrt {a+b x^2} (7 a d+2 b c)}{48 c^2 d \left (c+d x^2\right )^3}-\frac {x \sqrt {a+b x^2} (b c-a d)}{8 c d \left (c+d x^2\right )^4} \]

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Rubi [A]  time = 0.37, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {413, 527, 12, 377, 208} \begin {gather*} \frac {x \sqrt {a+b x^2} \left (-170 a^2 b c d^2+105 a^3 d^3+40 a b^2 c^2 d+16 b^3 c^3\right )}{384 c^4 d \left (c+d x^2\right ) (b c-a d)^2}+\frac {x \sqrt {a+b x^2} \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{192 c^3 d \left (c+d x^2\right )^2 (b c-a d)}+\frac {a^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \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)^{5/2}}+\frac {x \sqrt {a+b x^2} (7 a d+2 b c)}{48 c^2 d \left (c+d x^2\right )^3}-\frac {x \sqrt {a+b x^2} (b c-a d)}{8 c d \left (c+d x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]

[Out]

-((b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c*d*(c + d*x^2)^4) + ((2*b*c + 7*a*d)*x*Sqrt[a + b*x^2])/(48*c^2*d*(c + d*
x^2)^3) + ((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2])/(192*c^3*d*(b*c - a*d)*(c + d*x^2)^2) + ((
16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(384*c^4*d*(b*c - a*d)^2*(c +
d*x^2)) + (a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/
(128*c^(9/2)*(b*c - a*d)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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 413

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

Rule 527

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^5} \, dx &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {\int \frac {a (b c+7 a d)+2 b (b c+3 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^4} \, dx}{8 c d}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\int \frac {a (b c-a d) (4 b c+35 a d)+4 b (b c-a d) (2 b c+7 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx}{48 c^2 d (b c-a d)}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2}}{192 c^3 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\int \frac {a (b c-a d) \left (8 b^2 c^2+100 a b c d-105 a^2 d^2\right )+2 b (b c-a d) \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{192 c^3 d (b c-a d)^2}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2}}{192 c^3 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{384 c^4 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\int \frac {3 a^2 d (b c-a d) \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{384 c^4 d (b c-a d)^3}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2}}{192 c^3 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{384 c^4 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{128 c^4 (b c-a d)^2}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2}}{192 c^3 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{384 c^4 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\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)^2}\\ &=-\frac {(b c-a d) x \sqrt {a+b x^2}}{8 c d \left (c+d x^2\right )^4}+\frac {(2 b c+7 a d) x \sqrt {a+b x^2}}{48 c^2 d \left (c+d x^2\right )^3}+\frac {\left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2}}{192 c^3 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{384 c^4 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \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)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 1.38, size = 362, normalized size = 1.21 \begin {gather*} \frac {a x \left (\frac {b x^2}{a}+1\right ) \left (\frac {3 a^2 \left (c+d x^2\right )^4 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \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 (a^4 d^2 \left (279 c^3+511 c^2 d x^2+385 c d^2 x^4+105 d^3 x^6\right )+a^3 b d \left (-528 c^4-563 c^3 d x^2-117 c^2 d^2 x^4+215 c d^3 x^6+105 d^4 x^8\right )+2 a^2 b^2 c \left (120 c^4-160 c^3 d x^2-345 c^2 d^2 x^4-294 c d^3 x^6-85 d^4 x^8\right )+8 a b^3 c^2 x^2 \left (42 c^3+34 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )+16 b^4 c^3 x^4 \left (6 c^2+4 c d x^2+d^2 x^4\right )\right )\right )}{384 c^5 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^4 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]

[Out]

(a*x*(1 + (b*x^2)/a)*(c*(16*b^4*c^3*x^4*(6*c^2 + 4*c*d*x^2 + d^2*x^4) + 8*a*b^3*c^2*x^2*(42*c^3 + 34*c^2*d*x^2
+ 21*c*d^2*x^4 + 5*d^3*x^6) + a^4*d^2*(279*c^3 + 511*c^2*d*x^2 + 385*c*d^2*x^4 + 105*d^3*x^6) + 2*a^2*b^2*c*(
120*c^4 - 160*c^3*d*x^2 - 345*c^2*d^2*x^4 - 294*c*d^3*x^6 - 85*d^4*x^8) + a^3*b*d*(-528*c^4 - 563*c^3*d*x^2 -
117*c^2*d^2*x^4 + 215*c*d^3*x^6 + 105*d^4*x^8)) + (3*a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*(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)^2*(a + b*x^2)^(3/2)*(c + d*x^2)^4)

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IntegrateAlgebraic [B]  time = 43.48, size = 2592, normalized size = 8.64 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]

[Out]

(Sqrt[a + b*x^2]*(279*a^7*c^3*d^4*x + 511*a^7*c^2*d^5*x^3 + 385*a^7*c*d^6*x^5 + 105*a^7*d^7*x^7) + Sqrt[b]*(10
5*a^7*c^4*d^3 - 1812*a^7*c^3*d^4*x^2 - 3458*a^7*c^2*d^5*x^4 - 2660*a^7*c*d^6*x^6 - 735*a^7*d^7*x^8) + b^7*Sqrt
[a + b*x^2]*(-2048*c^7*x^7 - 8192*c^6*d*x^9) + b^6*Sqrt[a + b*x^2]*(-3072*a*c^7*x^5 - 17408*a*c^6*d*x^7 + 1638
4*a*c^5*d^2*x^9) + b*Sqrt[a + b*x^2]*(-1368*a^6*c^4*d^3*x + 4726*a^6*c^3*d^4*x^3 + 10684*a^6*c^2*d^5*x^5 + 879
0*a^6*c*d^6*x^7 + 2520*a^6*d^7*x^9) + b^(15/2)*(2048*c^7*x^8 + 8192*c^6*d*x^10) + b^(13/2)*(4096*a*c^7*x^6 + 2
1504*a*c^6*d*x^8 - 16384*a*c^5*d^2*x^10) + b^(3/2)*(-170*a^6*c^5*d^2 + 6904*a^6*c^4*d^3*x^2 - 5396*a^6*c^3*d^4
*x^4 - 20464*a^6*c^2*d^5*x^6 - 19250*a^6*c*d^6*x^8 - 5880*a^6*d^7*x^10) + b^2*Sqrt[a + b*x^2]*(1600*a^5*c^5*d^
2*x - 19648*a^5*c^4*d^3*x^3 - 7592*a^5*c^3*d^4*x^5 + 16744*a^5*c^2*d^5*x^7 + 23920*a^5*c*d^6*x^9 + 8400*a^5*d^
7*x^11) + b^(5/2)*(40*a^5*c^6*d - 7200*a^5*c^5*d^2*x^2 + 40080*a^5*c^4*d^3*x^4 + 31712*a^5*c^3*d^4*x^6 - 4968*
a^5*c^2*d^5*x^8 - 28000*a^5*c*d^6*x^10 - 11760*a^5*d^7*x^12) + b^5*Sqrt[a + b*x^2]*(-1280*a^2*c^7*x^3 - 12800*
a^2*c^6*d*x^5 + 60160*a^2*c^5*d^2*x^7 + 31744*a^2*c^4*d^3*x^9 + 32256*a^2*c^3*d^4*x^11 + 9216*a^2*c^2*d^5*x^13
) + b^4*Sqrt[a + b*x^2]*(-128*a^3*c^7*x - 3712*a^3*c^6*d*x^3 + 60544*a^3*c^5*d^2*x^5 - 2432*a^3*c^4*d^3*x^7 -
25856*a^3*c^3*d^4*x^9 - 42240*a^3*c^2*d^5*x^11 - 15360*a^3*c*d^6*x^13) + b^3*Sqrt[a + b*x^2]*(-320*a^4*c^6*d*x
+ 20096*a^4*c^5*d^2*x^3 - 45440*a^4*c^4*d^3*x^5 - 58736*a^4*c^3*d^4*x^7 - 35264*a^4*c^2*d^5*x^9 + 4320*a^4*c*
d^6*x^11 + 6720*a^4*d^7*x^13) + b^(11/2)*(2560*a^2*c^7*x^4 + 20480*a^2*c^6*d*x^6 - 68352*a^2*c^5*d^2*x^8 - 317
44*a^2*c^4*d^3*x^10 - 32256*a^2*c^3*d^4*x^12 - 9216*a^2*c^2*d^5*x^14) + b^(9/2)*(512*a^3*c^7*x^2 + 8448*a^3*c^
6*d*x^4 - 88576*a^3*c^5*d^2*x^6 - 13440*a^3*c^4*d^3*x^8 + 9728*a^3*c^3*d^4*x^10 + 37632*a^3*c^2*d^5*x^12 + 153
60*a^3*c*d^6*x^14) + b^(7/2)*(16*a^4*c^7 + 1344*a^4*c^6*d*x^2 - 43872*a^4*c^5*d^2*x^4 + 50624*a^4*c^4*d^3*x^6
+ 75696*a^4*c^3*d^4*x^8 + 57536*a^4*c^2*d^5*x^10 + 3360*a^4*c*d^6*x^12 - 6720*a^4*d^7*x^14))/(384*a^6*c^4*d^4*
(c + d*x^2)^4 + 49152*b^6*c^6*d^2*x^8*(c + d*x^2)^4 - 3072*a^5*Sqrt[b]*c^4*d^4*x*Sqrt[a + b*x^2]*(c + d*x^2)^4
- 49152*b^(11/2)*c^6*d^2*x^7*Sqrt[a + b*x^2]*(c + d*x^2)^4 + 384*b*c^4*d^2*(c + d*x^2)^4*(-2*a^5*c*d + 32*a^5
*d^2*x^2) + 384*b^(3/2)*c^4*d^2*Sqrt[a + b*x^2]*(c + d*x^2)^4*(16*a^4*c*d*x - 80*a^4*d^2*x^3) + 384*b^2*c^4*d^
2*(c + d*x^2)^4*(a^4*c^2 - 64*a^4*c*d*x^2 + 160*a^4*d^2*x^4) + 384*b^(5/2)*c^4*d^2*Sqrt[a + b*x^2]*(c + d*x^2)
^4*(-8*a^3*c^2*x + 160*a^3*c*d*x^3 - 192*a^3*d^2*x^5) + 384*b^3*c^4*d^2*(c + d*x^2)^4*(32*a^3*c^2*x^2 - 320*a^
3*c*d*x^4 + 256*a^3*d^2*x^6) + 384*b^(9/2)*c^4*d^2*Sqrt[a + b*x^2]*(c + d*x^2)^4*(-192*a*c^2*x^5 + 256*a*c*d*x
^7) + 384*b^(7/2)*c^4*d^2*Sqrt[a + b*x^2]*(c + d*x^2)^4*(-80*a^2*c^2*x^3 + 384*a^2*c*d*x^5 - 128*a^2*d^2*x^7)
+ 384*b^5*c^4*d^2*(c + d*x^2)^4*(256*a*c^2*x^6 - 256*a*c*d*x^8) + 384*b^4*c^4*d^2*(c + d*x^2)^4*(160*a^2*c^2*x
^4 - 512*a^2*c*d*x^6 + 128*a^2*d^2*x^8)) + (2*b^4*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b*c) + a*d] + (Sqrt[b]*d*x^2
)/(Sqrt[c]*Sqrt[-(b*c) + a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(Sqrt[c]*d^2*(b*c - a*d)
^2*Sqrt[-(b*c) + a*d]) - (2*a*b^3*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b*c) + a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[
-(b*c) + a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(c^(3/2)*d*(b*c - a*d)^2*Sqrt[-(b*c) + a
*d]) + (3*a^2*b^2*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x
*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(8*c^(5/2)*(b*c - a*d)^(5/2)) + (2*b^4*ArcTanh[(Sqrt[b]*Sqrt[c])
/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])
])/(Sqrt[c]*d^2*(b*c - a*d)^(5/2)) - (2*a*b^3*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqr
t[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(c^(3/2)*d*(b*c - a*d)^(5/2)) - (5*a
^3*b*d*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b
*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(8*c^(7/2)*(b*c - a*d)^(5/2)) + (35*a^4*d^2*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[
b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(12
8*c^(9/2)*(b*c - a*d)^(5/2))

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fricas [B]  time = 5.36, size = 1604, normalized size = 5.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^5,x, algorithm="fricas")

[Out]

[1/1536*(3*(48*a^2*b^2*c^6 - 80*a^3*b*c^5*d + 35*a^4*c^4*d^2 + (48*a^2*b^2*c^2*d^4 - 80*a^3*b*c*d^5 + 35*a^4*d
^6)*x^8 + 4*(48*a^2*b^2*c^3*d^3 - 80*a^3*b*c^2*d^4 + 35*a^4*c*d^5)*x^6 + 6*(48*a^2*b^2*c^4*d^2 - 80*a^3*b*c^3*
d^3 + 35*a^4*c^2*d^4)*x^4 + 4*(48*a^2*b^2*c^5*d - 80*a^3*b*c^4*d^2 + 35*a^4*c^3*d^3)*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^2 + 24*a*b^3*c^4*d
^3 - 210*a^2*b^2*c^3*d^4 + 275*a^3*b*c^2*d^5 - 105*a^4*c*d^6)*x^7 + (64*b^4*c^6*d + 88*a*b^3*c^5*d^2 - 780*a^2
*b^2*c^4*d^3 + 1013*a^3*b*c^3*d^4 - 385*a^4*c^2*d^5)*x^5 + (96*b^4*c^7 + 112*a*b^3*c^6*d - 1050*a^2*b^2*c^5*d^
2 + 1353*a^3*b*c^4*d^3 - 511*a^4*c^3*d^4)*x^3 + 3*(80*a*b^3*c^7 - 256*a^2*b^2*c^6*d + 269*a^3*b*c^5*d^2 - 93*a
^4*c^4*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^12 - 3*a*b^2*c^11*d + 3*a^2*b*c^10*d^2 - a^3*c^9*d^3 + (b^3*c^8*d^4 - 3
*a*b^2*c^7*d^5 + 3*a^2*b*c^6*d^6 - a^3*c^5*d^7)*x^8 + 4*(b^3*c^9*d^3 - 3*a*b^2*c^8*d^4 + 3*a^2*b*c^7*d^5 - a^3
*c^6*d^6)*x^6 + 6*(b^3*c^10*d^2 - 3*a*b^2*c^9*d^3 + 3*a^2*b*c^8*d^4 - a^3*c^7*d^5)*x^4 + 4*(b^3*c^11*d - 3*a*b
^2*c^10*d^2 + 3*a^2*b*c^9*d^3 - a^3*c^8*d^4)*x^2), -1/768*(3*(48*a^2*b^2*c^6 - 80*a^3*b*c^5*d + 35*a^4*c^4*d^2
+ (48*a^2*b^2*c^2*d^4 - 80*a^3*b*c*d^5 + 35*a^4*d^6)*x^8 + 4*(48*a^2*b^2*c^3*d^3 - 80*a^3*b*c^2*d^4 + 35*a^4*
c*d^5)*x^6 + 6*(48*a^2*b^2*c^4*d^2 - 80*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^4 + 4*(48*a^2*b^2*c^5*d - 80*a^3*b*c
^4*d^2 + 35*a^4*c^3*d^3)*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)*s
qrt(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^2 + 24*a*b^3*c^4*d^3 - 21
0*a^2*b^2*c^3*d^4 + 275*a^3*b*c^2*d^5 - 105*a^4*c*d^6)*x^7 + (64*b^4*c^6*d + 88*a*b^3*c^5*d^2 - 780*a^2*b^2*c^
4*d^3 + 1013*a^3*b*c^3*d^4 - 385*a^4*c^2*d^5)*x^5 + (96*b^4*c^7 + 112*a*b^3*c^6*d - 1050*a^2*b^2*c^5*d^2 + 135
3*a^3*b*c^4*d^3 - 511*a^4*c^3*d^4)*x^3 + 3*(80*a*b^3*c^7 - 256*a^2*b^2*c^6*d + 269*a^3*b*c^5*d^2 - 93*a^4*c^4*
d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^12 - 3*a*b^2*c^11*d + 3*a^2*b*c^10*d^2 - a^3*c^9*d^3 + (b^3*c^8*d^4 - 3*a*b^2*
c^7*d^5 + 3*a^2*b*c^6*d^6 - a^3*c^5*d^7)*x^8 + 4*(b^3*c^9*d^3 - 3*a*b^2*c^8*d^4 + 3*a^2*b*c^7*d^5 - a^3*c^6*d^
6)*x^6 + 6*(b^3*c^10*d^2 - 3*a*b^2*c^9*d^3 + 3*a^2*b*c^8*d^4 - a^3*c^7*d^5)*x^4 + 4*(b^3*c^11*d - 3*a*b^2*c^10
*d^2 + 3*a^2*b*c^9*d^3 - a^3*c^8*d^4)*x^2)]

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giac [B]  time = 9.60, size = 1557, normalized size = 5.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^5,x, algorithm="giac")

[Out]

-1/128*(48*a^2*b^(5/2)*c^2 - 80*a^3*b^(3/2)*c*d + 35*a^4*sqrt(b)*d^2)*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^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2)*sqrt(-b^2*c^2 + a*b*c*d)
) - 1/192*(144*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(5/2)*c^2*d^5 - 240*(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 + 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^12*
a^2*b^(7/2)*c^3*d^4 - 4368*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(5/2)*c^2*d^5 + 3150*(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 + 4096*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(11/2)*c^5*d^2 + 7936*(sqrt(b)*x - sqr
t(b*x^2 + a))^10*a^2*b^(9/2)*c^4*d^3 - 26624*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(7/2)*c^3*d^4 + 26944*(sqr
t(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(5/2)*c^2*d^5 - 12320*(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 - 10
24*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(13/2)*c^6*d + 27392*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(11/2)*c^5*d
^2 - 65920*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(9/2)*c^4*d^3 + 81680*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(
7/2)*c^3*d^4 - 58840*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(5/2)*c^2*d^5 + 22750*(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 - 8192*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(11/2)*c^5*d^2 + 47104*(sqrt(b)*x - sqrt(
b*x^2 + a))^6*a^4*b^(9/2)*c^4*d^3 - 74240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(7/2)*c^3*d^4 + 56416*(sqrt(b)
*x - sqrt(b*x^2 + a))^6*a^6*b^(5/2)*c^2*d^5 - 22400*(sqrt(b)*x - sqrt(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 - 1536*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(11/2)*c^5*d^2 - 2
304*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(9/2)*c^4*d^3 + 17696*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*b^(7/2)*c^
3*d^4 - 23152*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(5/2)*c^2*d^5 + 11690*(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 - 512*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(7/2)*c^3*d^4 + 2896*(sqrt(b)*x - sqrt(b*x^2 + a)
)^2*a^8*b^(5/2)*c^2*d^5 - 2800*(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 - 40*a^9*b^(5/2)*c^2*d^5 + 170*a^10*b^(3/2)*c*d^6 - 105*a^1
1*sqrt(b)*d^7)/((b^2*c^6*d^2 - 2*a*b*c^5*d^3 + a^2*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)

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maple [B]  time = 0.05, size = 18791, normalized size = 62.64 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/(d*x^2+c)^5,x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^5,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(3/2)/(c + d*x^2)^5,x)

[Out]

int((a + b*x^2)^(3/2)/(c + d*x^2)^5, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**5,x)

[Out]

Timed out

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