3.1.70 \(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^5} \, dx\) [70]

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

[Out]

-1/8*d*x*(b*x^2+a)^(7/2)/c/(-a*d+b*c)/(d*x^2+c)^4+1/48*(-7*a*d+8*b*c)*x*(b*x^2+a)^(5/2)/c^2/(-a*d+b*c)/(d*x^2+
c)^3+5/192*a*(-7*a*d+8*b*c)*x*(b*x^2+a)^(3/2)/c^3/(-a*d+b*c)/(d*x^2+c)^2+5/128*a^3*(-7*a*d+8*b*c)*arctanh(x*(-
a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(9/2)/(-a*d+b*c)^(3/2)+5/128*a^2*(-7*a*d+8*b*c)*x*(b*x^2+a)^(1/2)/c^
4/(-a*d+b*c)/(d*x^2+c)

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Rubi [A]
time = 0.09, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {390, 386, 385, 214} \begin {gather*} \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 {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 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 {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 {d x \left (a+b x^2\right )^{7/2}}{8 c \left (c+d x^2\right )^4 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(d*x*(a + b*x^2)^(7/2))/(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 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 386

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 390

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] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 ) \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)}\\ &=-\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*}

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Mathematica [A]
time = 10.75, size = 306, normalized size = 1.23 \begin {gather*} \frac {x \left (-c \left (16 b^4 c^3 x^6 \left (4 c+d x^2\right )+8 a b^3 c^2 x^4 \left (34 c^2+13 c d x^2+3 d^2 x^4\right )+2 a^2 b^2 c x^2 \left (236 c^3+173 c^2 d x^2+106 c d^2 x^4+25 d^3 x^6\right )-a^4 d \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 \left (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\right )\right )+\frac {15 a^3 (-8 b c+7 a d) \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) \sqrt {a+b x^2} \left (c+d x^2\right )^4} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(34026\) vs. \(2(221)=442\).
time = 0.08, size = 34027, normalized size = 136.65

method	result	size
default	\(\text {Expression too large to display}\)	\(34027\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/(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)^(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] Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (221) = 442\).
time = 1.03, size = 1258, normalized size = 5.05 \begin {gather*} \left [\frac {15 \, {\left (8 \, a^{3} b c^{5} - 7 \, a^{4} c^{4} d + {\left (8 \, a^{3} b c d^{4} - 7 \, a^{4} d^{5}\right )} x^{8} + 4 \, {\left (8 \, a^{3} b c^{2} d^{3} - 7 \, a^{4} c d^{4}\right )} x^{6} + 6 \, {\left (8 \, a^{3} b c^{3} d^{2} - 7 \, a^{4} c^{2} d^{3}\right )} x^{4} + 4 \, {\left (8 \, a^{3} b c^{4} d - 7 \, a^{4} c^{3} d^{2}\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 + 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}\right )} x^{7} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c^{6} - 181 \, a^{3} b c^{5} d + 93 \, a^{4} c^{4} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{1536 \, {\left (b^{2} c^{11} - 2 \, a b c^{10} d + a^{2} c^{9} d^{2} + {\left (b^{2} c^{7} d^{4} - 2 \, a b c^{6} d^{5} + a^{2} c^{5} d^{6}\right )} x^{8} + 4 \, {\left (b^{2} c^{8} d^{3} - 2 \, a b c^{7} d^{4} + a^{2} c^{6} d^{5}\right )} x^{6} + 6 \, {\left (b^{2} c^{9} d^{2} - 2 \, a b c^{8} d^{3} + a^{2} c^{7} d^{4}\right )} x^{4} + 4 \, {\left (b^{2} c^{10} d - 2 \, a b c^{9} d^{2} + a^{2} c^{8} d^{3}\right )} x^{2}\right )}}, -\frac {15 \, {\left (8 \, a^{3} b c^{5} - 7 \, a^{4} c^{4} d + {\left (8 \, a^{3} b c d^{4} - 7 \, a^{4} d^{5}\right )} x^{8} + 4 \, {\left (8 \, a^{3} b c^{2} d^{3} - 7 \, a^{4} c d^{4}\right )} x^{6} + 6 \, {\left (8 \, a^{3} b c^{3} d^{2} - 7 \, a^{4} c^{2} d^{3}\right )} x^{4} + 4 \, {\left (8 \, a^{3} b c^{4} d - 7 \, a^{4} c^{3} d^{2}\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 + 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}\right )} x^{7} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c^{6} - 181 \, a^{3} b c^{5} d + 93 \, a^{4} c^{4} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, {\left (b^{2} c^{11} - 2 \, a b c^{10} d + a^{2} c^{9} d^{2} + {\left (b^{2} c^{7} d^{4} - 2 \, a b c^{6} d^{5} + a^{2} c^{5} d^{6}\right )} x^{8} + 4 \, {\left (b^{2} c^{8} d^{3} - 2 \, a b c^{7} d^{4} + a^{2} c^{6} d^{5}\right )} x^{6} + 6 \, {\left (b^{2} c^{9} d^{2} - 2 \, a b c^{8} d^{3} + a^{2} c^{7} d^{4}\right )} x^{4} + 4 \, {\left (b^{2} c^{10} d - 2 \, a b c^{9} d^{2} + a^{2} c^{8} d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}

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)] 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)**(5/2)/(d*x**2+c)**5,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (221) = 442\).
time = 5.21, size = 1448, normalized size = 5.82 \begin {gather*} -\frac {5 \, {\left (8 \, a^{3} b^{\frac {3}{2}} c - 7 \, a^{4} \sqrt {b} d\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 c^{5} - a c^{4} d\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {120 \, {\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} - 768 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {11}{2}} c^{5} d^{2} + 768 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a b^{\frac {9}{2}} c^{4} d^{3} + 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {5}{2}} c^{2} d^{5} - 2310 \, {\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 + 2048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {9}{2}} c^{4} d^{3} + 8320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} b^{\frac {7}{2}} c^{3} d^{4} - 15600 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {5}{2}} c^{2} d^{5} + 9800 \, {\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 - 4864 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {11}{2}} c^{5} d^{2} + 21888 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {9}{2}} c^{4} d^{3} - 38000 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {7}{2}} c^{3} d^{4} + 37400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {5}{2}} c^{2} d^{5} - 18550 \, {\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 - 9472 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {9}{2}} c^{4} d^{3} + 32896 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {7}{2}} c^{3} d^{4} - 35376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {5}{2}} c^{2} d^{5} + 18200 \, {\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} - 768 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {11}{2}} c^{5} d^{2} - 1536 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {9}{2}} c^{4} d^{3} - 2944 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {7}{2}} c^{3} d^{4} + 12528 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {5}{2}} c^{2} d^{5} - 9170 \, {\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} - 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {7}{2}} c^{3} d^{4} - 608 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {5}{2}} c^{2} d^{5} + 1960 \, {\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} - 24 \, a^{9} b^{\frac {5}{2}} c^{2} d^{5} - 50 \, a^{10} b^{\frac {3}{2}} c d^{6} + 105 \, a^{11} \sqrt {b} d^{7}}{192 \, {\left (b c^{5} d^{3} - a 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)^(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)

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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 )}^{5/2}}{{\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)^(5/2)/(c + d*x^2)^5,x)

[Out]

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

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