3.1.61 \(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^5} \, dx\) [61]

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

[Out]

1/128*a^2*(35*a^2*d^2-80*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^(9/2)/(-a*d
+b*c)^(5/2)-1/8*(-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/d/(d*x^2+c)^4+1/48*(7*a*d+2*b*c)*x*(b*x^2+a)^(1/2)/c^2/d/(d*x^2
+c)^3+1/192*(-35*a^2*d^2+24*a*b*c*d+8*b^2*c^2)*x*(b*x^2+a)^(1/2)/c^3/d/(-a*d+b*c)/(d*x^2+c)^2+1/384*(105*a^3*d
^3-170*a^2*b*c*d^2+40*a*b^2*c^2*d+16*b^3*c^3)*x*(b*x^2+a)^(1/2)/c^4/d/(-a*d+b*c)^2/(d*x^2+c)

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Rubi [A]
time = 0.25, 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 = {424, 541, 12, 385, 214} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((b*c - a*d)*x*Sqrt[a + b*x^2])/(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 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 424

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 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 {\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 ) \text {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 = 11.02, size = 362, normalized size = 1.21 \begin {gather*} \frac {a x \left (1+\frac {b x^2}{a}\right ) \left (c \left (16 b^4 c^3 x^4 \left (6 c^2+4 c d x^2+d^2 x^4\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 )+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 )+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 )+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 )\right )+\frac {3 a^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \left (c+d x^2\right )^4 \tanh ^{-1}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{384 c^5 (b c-a d)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^4} \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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(22501\) vs. \(2(272)=544\).
time = 0.09, size = 22502, normalized size = 75.01

method	result	size
default	\(\text {Expression too large to display}\)	\(22502\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/(d*x^2+c)^5,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((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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (272) = 544\).
time = 1.76, size = 1604, normalized size = 5.35 \begin {gather*} \left [\frac {3 \, {\left (48 \, a^{2} b^{2} c^{6} - 80 \, a^{3} b c^{5} d + 35 \, a^{4} c^{4} d^{2} + {\left (48 \, a^{2} b^{2} c^{2} d^{4} - 80 \, a^{3} b c d^{5} + 35 \, a^{4} d^{6}\right )} x^{8} + 4 \, {\left (48 \, a^{2} b^{2} c^{3} d^{3} - 80 \, a^{3} b c^{2} d^{4} + 35 \, a^{4} c d^{5}\right )} x^{6} + 6 \, {\left (48 \, a^{2} b^{2} c^{4} d^{2} - 80 \, a^{3} b c^{3} d^{3} + 35 \, a^{4} c^{2} d^{4}\right )} x^{4} + 4 \, {\left (48 \, a^{2} b^{2} c^{5} d - 80 \, a^{3} b c^{4} d^{2} + 35 \, a^{4} c^{3} d^{3}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (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}\right )} x^{7} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{3} + 3 \, {\left (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}\right )} x\right )} \sqrt {b x^{2} + a}}{1536 \, {\left (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} + {\left (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}\right )} x^{8} + 4 \, {\left (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}\right )} x^{6} + 6 \, {\left (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}\right )} x^{4} + 4 \, {\left (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}\right )} x^{2}\right )}}, -\frac {3 \, {\left (48 \, a^{2} b^{2} c^{6} - 80 \, a^{3} b c^{5} d + 35 \, a^{4} c^{4} d^{2} + {\left (48 \, a^{2} b^{2} c^{2} d^{4} - 80 \, a^{3} b c d^{5} + 35 \, a^{4} d^{6}\right )} x^{8} + 4 \, {\left (48 \, a^{2} b^{2} c^{3} d^{3} - 80 \, a^{3} b c^{2} d^{4} + 35 \, a^{4} c d^{5}\right )} x^{6} + 6 \, {\left (48 \, a^{2} b^{2} c^{4} d^{2} - 80 \, a^{3} b c^{3} d^{3} + 35 \, a^{4} c^{2} d^{4}\right )} x^{4} + 4 \, {\left (48 \, a^{2} b^{2} c^{5} d - 80 \, a^{3} b c^{4} d^{2} + 35 \, a^{4} c^{3} d^{3}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (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}\right )} x^{7} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{3} + 3 \, {\left (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}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, {\left (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} + {\left (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}\right )} x^{8} + 4 \, {\left (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}\right )} x^{6} + 6 \, {\left (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}\right )} x^{4} + 4 \, {\left (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}\right )} x^{2}\right )}}\right ] \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="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)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1557 vs. \(2 (272) = 544\).
time = 5.11, size = 1557, normalized size = 5.19 \begin {gather*} -\frac {{\left (48 \, a^{2} b^{\frac {5}{2}} c^{2} - 80 \, a^{3} b^{\frac {3}{2}} c d + 35 \, a^{4} \sqrt {b} d^{2}\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 )}{128 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {144 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{2} b^{\frac {5}{2}} c^{2} d^{5} - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{3} b^{\frac {3}{2}} c d^{6} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} \sqrt {b} d^{7} + 2016 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{2} b^{\frac {7}{2}} c^{3} d^{4} - 4368 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {5}{2}} c^{2} d^{5} + 3150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{4} b^{\frac {3}{2}} c d^{6} - 735 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} \sqrt {b} d^{7} - 2048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {13}{2}} c^{6} d + 4096 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {11}{2}} c^{5} d^{2} + 7936 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {9}{2}} c^{4} d^{3} - 26624 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} b^{\frac {7}{2}} c^{3} d^{4} + 26944 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {5}{2}} c^{2} d^{5} - 12320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{5} b^{\frac {3}{2}} c d^{6} + 2205 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} \sqrt {b} d^{7} - 2048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {15}{2}} c^{7} - 1024 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {13}{2}} c^{6} d + 27392 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {11}{2}} c^{5} d^{2} - 65920 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {9}{2}} c^{4} d^{3} + 81680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {7}{2}} c^{3} d^{4} - 58840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {5}{2}} c^{2} d^{5} + 22750 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{6} b^{\frac {3}{2}} c d^{6} - 3675 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} \sqrt {b} d^{7} - 2048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {13}{2}} c^{6} d - 8192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {11}{2}} c^{5} d^{2} + 47104 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {9}{2}} c^{4} d^{3} - 74240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {7}{2}} c^{3} d^{4} + 56416 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {5}{2}} c^{2} d^{5} - 22400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{7} b^{\frac {3}{2}} c d^{6} + 3675 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} \sqrt {b} d^{7} - 1536 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {11}{2}} c^{5} d^{2} - 2304 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {9}{2}} c^{4} d^{3} + 17696 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {7}{2}} c^{3} d^{4} - 23152 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {5}{2}} c^{2} d^{5} + 11690 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{8} b^{\frac {3}{2}} c d^{6} - 2205 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} \sqrt {b} d^{7} - 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {9}{2}} c^{4} d^{3} - 512 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {7}{2}} c^{3} d^{4} + 2896 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {5}{2}} c^{2} d^{5} - 2800 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{9} b^{\frac {3}{2}} c d^{6} + 735 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} \sqrt {b} d^{7} - 16 \, a^{8} b^{\frac {7}{2}} c^{3} d^{4} - 40 \, a^{9} b^{\frac {5}{2}} c^{2} d^{5} + 170 \, a^{10} b^{\frac {3}{2}} c d^{6} - 105 \, a^{11} \sqrt {b} d^{7}}{192 \, {\left (b^{2} c^{6} d^{2} - 2 \, a b c^{5} d^{3} + a^{2} c^{4} 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 )}^{4}} \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="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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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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