3.1.96 \(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^3} \, dx\) [96]

Optimal. Leaf size=313 \[ -\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}} \]

[Out]

-1/4*d*x/c/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^2+1/12*b*(3*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^2+a)^(3/2)/(d*x
^2+c)+1/8*d^2*(3*a^2*d^2-16*a*b*c*d+48*b^2*c^2)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(5/2)/(-
a*d+b*c)^(9/2)+1/12*b*(-3*a^2*d^2-40*a*b*c*d+8*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(d*x^2+c)/(b*x^2+a)^(1/2)+1/24*d*
(9*a^3*d^3-42*a^2*b*c*d^2-88*a*b^2*c^2*d+16*b^3*c^3)*x*(b*x^2+a)^(1/2)/a^2/c^2/(-a*d+b*c)^4/(d*x^2+c)

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Rubi [A]
time = 0.29, antiderivative size = 313, 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 = {425, 541, 12, 385, 214} \begin {gather*} \frac {b x \left (-3 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{12 a^2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)^3}+\frac {d^2 \left (3 a^2 d^2-16 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 )}{8 c^{5/2} (b c-a d)^{9/2}}+\frac {d x \sqrt {a+b x^2} \left (9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}-\frac {d x}{4 c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (b c-a d)}+\frac {b x (3 a d+4 b c)}{12 a c \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]

[Out]

-1/4*(d*x)/(c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + (b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a +
b*x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]
*(c + d*x^2)) + (d*(16*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a^2*c^2*(
b*c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sq
rt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(9/2))

Rule 12

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

Rule 214

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

Rule 385

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 425

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[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

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 {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3} \, dx &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-6 b d x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}-\frac {\int \frac {-8 b^2 c^2+24 a b c d-9 a^2 d^2-4 b d (4 b c+3 a d) x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx}{12 a c (b c-a d)^2}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {a d \left (8 b^2 c^2+36 a b c d-9 a^2 d^2\right )+2 b d \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{12 a^2 c (b c-a d)^3}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\int \frac {3 a^2 d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{24 a^2 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\left (d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^4}\\ &=-\frac {d x}{4 c (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}+\frac {b (4 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}+\frac {b \left (8 b^2 c^2-40 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d \left (16 b^3 c^3-88 a b^2 c^2 d-42 a^2 b c d^2+9 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {d^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 3.78, size = 367, normalized size = 1.17 \begin {gather*} \frac {\frac {\sqrt {c} x \left (16 b^5 c^3 x^2 \left (c+d x^2\right )^2+8 a b^4 c^2 \left (3 c-11 d x^2\right ) \left (c+d x^2\right )^2+3 a^5 d^4 \left (5 c+3 d x^2\right )+3 a^3 b^2 d^3 x^2 \left (-32 c^2-23 c d x^2+3 d^2 x^4\right )+6 a^4 b d^3 \left (-8 c^2-2 c d x^2+3 d^2 x^4\right )-6 a^2 b^3 c d \left (16 c^3+32 c^2 d x^2+24 c d^2 x^4+7 d^3 x^6\right )\right )}{a^2 (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}-\frac {9 d^2 (-4 b c+a d)^2 \tan ^{-1}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{9/2}}+\frac {24 a b c d^3 \tanh ^{-1}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {b c-a d}}\right )}{(b c-a d)^{9/2}}}{24 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]

[Out]

((Sqrt[c]*x*(16*b^5*c^3*x^2*(c + d*x^2)^2 + 8*a*b^4*c^2*(3*c - 11*d*x^2)*(c + d*x^2)^2 + 3*a^5*d^4*(5*c + 3*d*
x^2) + 3*a^3*b^2*d^3*x^2*(-32*c^2 - 23*c*d*x^2 + 3*d^2*x^4) + 6*a^4*b*d^3*(-8*c^2 - 2*c*d*x^2 + 3*d^2*x^4) - 6
*a^2*b^3*c*d*(16*c^3 + 32*c^2*d*x^2 + 24*c*d^2*x^4 + 7*d^3*x^6)))/(a^2*(b*c - a*d)^4*(a + b*x^2)^(3/2)*(c + d*
x^2)^2) - (9*d^2*(-4*b*c + a*d)^2*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) +
a*d])])/(-(b*c) + a*d)^(9/2) + (24*a*b*c*d^3*ArcTanh[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*
Sqrt[b*c - a*d])])/(b*c - a*d)^(9/2))/(24*c^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7120\) vs. \(2(285)=570\).
time = 0.09, size = 7121, normalized size = 22.75

method	result	size
default	\(\text {Expression too large to display}\)	\(7121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (285) = 570\).
time = 12.31, size = 2250, normalized size = 7.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/96*(3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + (48*a^2*b^4*c^2*d^4 - 16*a^3*b^3*c*d^5 + 3*a
^4*b^2*d^6)*x^8 + 2*(48*a^2*b^4*c^3*d^3 + 32*a^3*b^3*c^2*d^4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^2*b
^4*c^4*d^2 + 176*a^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a^5*b*c*d^5 + 3*a^6*d^6)*x^4 + 2*(48*a^3*b^3*c^4*d^2
+ 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^6*c*d^5)*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)*sq
rt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((16*b^6*c^5*d^2 - 104*a*b^5*c^4*d^3 + 46*a^2*b^4*c^3*d^4 + 51
*a^3*b^3*c^2*d^5 - 9*a^4*b^2*c*d^6)*x^7 + (32*b^6*c^6*d - 184*a*b^5*c^5*d^2 + 8*a^2*b^4*c^4*d^3 + 75*a^3*b^3*c
^3*d^4 + 87*a^4*b^2*c^2*d^5 - 18*a^5*b*c*d^6)*x^5 + (16*b^6*c^7 - 56*a*b^5*c^6*d - 152*a^2*b^4*c^5*d^2 + 96*a^
3*b^3*c^4*d^3 + 84*a^4*b^2*c^3*d^4 + 21*a^5*b*c^2*d^5 - 9*a^6*c*d^6)*x^3 + 3*(8*a*b^5*c^7 - 40*a^2*b^4*c^6*d +
32*a^3*b^3*c^5*d^2 - 16*a^4*b^2*c^4*d^3 + 21*a^5*b*c^3*d^4 - 5*a^6*c^2*d^5)*x)*sqrt(b*x^2 + a))/(a^4*b^5*c^10
- 5*a^5*b^4*c^9*d + 10*a^6*b^3*c^8*d^2 - 10*a^7*b^2*c^7*d^3 + 5*a^8*b*c^6*d^4 - a^9*c^5*d^5 + (a^2*b^7*c^8*d^
2 - 5*a^3*b^6*c^7*d^3 + 10*a^4*b^5*c^6*d^4 - 10*a^5*b^4*c^5*d^5 + 5*a^6*b^3*c^4*d^6 - a^7*b^2*c^3*d^7)*x^8 + 2
*(a^2*b^7*c^9*d - 4*a^3*b^6*c^8*d^2 + 5*a^4*b^5*c^7*d^3 - 5*a^6*b^3*c^5*d^5 + 4*a^7*b^2*c^4*d^6 - a^8*b*c^3*d^
7)*x^6 + (a^2*b^7*c^10 - a^3*b^6*c^9*d - 9*a^4*b^5*c^8*d^2 + 25*a^5*b^4*c^7*d^3 - 25*a^6*b^3*c^6*d^4 + 9*a^7*b
^2*c^5*d^5 + a^8*b*c^4*d^6 - a^9*c^3*d^7)*x^4 + 2*(a^3*b^6*c^10 - 4*a^4*b^5*c^9*d + 5*a^5*b^4*c^8*d^2 - 5*a^7*
b^2*c^6*d^4 + 4*a^8*b*c^5*d^5 - a^9*c^4*d^6)*x^2), -1/48*(3*(48*a^4*b^2*c^4*d^2 - 16*a^5*b*c^3*d^3 + 3*a^6*c^2
*d^4 + (48*a^2*b^4*c^2*d^4 - 16*a^3*b^3*c*d^5 + 3*a^4*b^2*d^6)*x^8 + 2*(48*a^2*b^4*c^3*d^3 + 32*a^3*b^3*c^2*d^
4 - 13*a^4*b^2*c*d^5 + 3*a^5*b*d^6)*x^6 + (48*a^2*b^4*c^4*d^2 + 176*a^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 - 4*a
^5*b*c*d^5 + 3*a^6*d^6)*x^4 + 2*(48*a^3*b^3*c^4*d^2 + 32*a^4*b^2*c^3*d^3 - 13*a^5*b*c^2*d^4 + 3*a^6*c*d^5)*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^6*c^5*d^2 - 104*a*b^5*c^4*d^3 + 46*a^2*b^4*c^3*d^4 + 51*a^3*b
^3*c^2*d^5 - 9*a^4*b^2*c*d^6)*x^7 + (32*b^6*c^6*d - 184*a*b^5*c^5*d^2 + 8*a^2*b^4*c^4*d^3 + 75*a^3*b^3*c^3*d^4
+ 87*a^4*b^2*c^2*d^5 - 18*a^5*b*c*d^6)*x^5 + (16*b^6*c^7 - 56*a*b^5*c^6*d - 152*a^2*b^4*c^5*d^2 + 96*a^3*b^3*
c^4*d^3 + 84*a^4*b^2*c^3*d^4 + 21*a^5*b*c^2*d^5 - 9*a^6*c*d^6)*x^3 + 3*(8*a*b^5*c^7 - 40*a^2*b^4*c^6*d + 32*a^
3*b^3*c^5*d^2 - 16*a^4*b^2*c^4*d^3 + 21*a^5*b*c^3*d^4 - 5*a^6*c^2*d^5)*x)*sqrt(b*x^2 + a))/(a^4*b^5*c^10 - 5*a
^5*b^4*c^9*d + 10*a^6*b^3*c^8*d^2 - 10*a^7*b^2*c^7*d^3 + 5*a^8*b*c^6*d^4 - a^9*c^5*d^5 + (a^2*b^7*c^8*d^2 - 5*
a^3*b^6*c^7*d^3 + 10*a^4*b^5*c^6*d^4 - 10*a^5*b^4*c^5*d^5 + 5*a^6*b^3*c^4*d^6 - a^7*b^2*c^3*d^7)*x^8 + 2*(a^2*
b^7*c^9*d - 4*a^3*b^6*c^8*d^2 + 5*a^4*b^5*c^7*d^3 - 5*a^6*b^3*c^5*d^5 + 4*a^7*b^2*c^4*d^6 - a^8*b*c^3*d^7)*x^6
+ (a^2*b^7*c^10 - a^3*b^6*c^9*d - 9*a^4*b^5*c^8*d^2 + 25*a^5*b^4*c^7*d^3 - 25*a^6*b^3*c^6*d^4 + 9*a^7*b^2*c^5
*d^5 + a^8*b*c^4*d^6 - a^9*c^3*d^7)*x^4 + 2*(a^3*b^6*c^10 - 4*a^4*b^5*c^9*d + 5*a^5*b^4*c^8*d^2 - 5*a^7*b^2*c^
6*d^4 + 4*a^8*b*c^5*d^5 - a^9*c^4*d^6)*x^2)]

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Sympy [F(-1)] Timed out
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(1/(b*x**2+a)**(5/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1010 vs. \(2 (285) = 570\).
time = 2.13, size = 1010, normalized size = 3.23 \begin {gather*} \frac {{\left (\frac {{\left (2 \, b^{10} c^{5} - 19 \, a b^{9} c^{4} d + 56 \, a^{2} b^{8} c^{3} d^{2} - 74 \, a^{3} b^{7} c^{2} d^{3} + 46 \, a^{4} b^{6} c d^{4} - 11 \, a^{5} b^{5} d^{5}\right )} x^{2}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}} + \frac {3 \, {\left (a b^{9} c^{5} - 8 \, a^{2} b^{8} c^{4} d + 22 \, a^{3} b^{7} c^{3} d^{2} - 28 \, a^{4} b^{6} c^{2} d^{3} + 17 \, a^{5} b^{5} c d^{4} - 4 \, a^{6} b^{4} d^{5}\right )}}{a^{2} b^{9} c^{8} - 8 \, a^{3} b^{8} c^{7} d + 28 \, a^{4} b^{7} c^{6} d^{2} - 56 \, a^{5} b^{6} c^{5} d^{3} + 70 \, a^{6} b^{5} c^{4} d^{4} - 56 \, a^{7} b^{4} c^{3} d^{5} + 28 \, a^{8} b^{3} c^{2} d^{6} - 8 \, a^{9} b^{2} c d^{7} + a^{10} b d^{8}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{\frac {5}{2}} c^{2} d^{2} - 16 \, a b^{\frac {3}{2}} c d^{3} + 3 \, a^{2} \sqrt {b} d^{4}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{4} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{5} + 112 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} d^{2} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d^{3} + 66 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{4} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{5} + 88 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d^{3} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{4} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{5} + 14 \, a^{4} b^{\frac {3}{2}} c d^{4} - 3 \, a^{5} \sqrt {b} d^{5}}{4 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/3*((2*b^10*c^5 - 19*a*b^9*c^4*d + 56*a^2*b^8*c^3*d^2 - 74*a^3*b^7*c^2*d^3 + 46*a^4*b^6*c*d^4 - 11*a^5*b^5*d^
5)*x^2/(a^2*b^9*c^8 - 8*a^3*b^8*c^7*d + 28*a^4*b^7*c^6*d^2 - 56*a^5*b^6*c^5*d^3 + 70*a^6*b^5*c^4*d^4 - 56*a^7*
b^4*c^3*d^5 + 28*a^8*b^3*c^2*d^6 - 8*a^9*b^2*c*d^7 + a^10*b*d^8) + 3*(a*b^9*c^5 - 8*a^2*b^8*c^4*d + 22*a^3*b^7
*c^3*d^2 - 28*a^4*b^6*c^2*d^3 + 17*a^5*b^5*c*d^4 - 4*a^6*b^4*d^5)/(a^2*b^9*c^8 - 8*a^3*b^8*c^7*d + 28*a^4*b^7*
c^6*d^2 - 56*a^5*b^6*c^5*d^3 + 70*a^6*b^5*c^4*d^4 - 56*a^7*b^4*c^3*d^5 + 28*a^8*b^3*c^2*d^6 - 8*a^9*b^2*c*d^7
+ a^10*b*d^8))*x/(b*x^2 + a)^(3/2) - 1/8*(48*b^(5/2)*c^2*d^2 - 16*a*b^(3/2)*c*d^3 + 3*a^2*sqrt(b)*d^4)*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^4*c^6 - 4*a*b^3*c^5*d + 6*
a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*d^4)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/4*(24*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*b^(5/2)*c^2*d^3 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d^4 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^
6*a^2*sqrt(b)*d^5 + 112*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*d^2 - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
a*b^(5/2)*c^2*d^3 + 66*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^4 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a
^3*sqrt(b)*d^5 + 88*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d^3 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a
^3*b^(3/2)*c*d^4 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^5 + 14*a^4*b^(3/2)*c*d^4 - 3*a^5*sqrt(b)*d^
5)/((b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*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)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x)

[Out]

int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3), x)

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