3.1.68 \(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2)^3} \, dx\) [68]

Optimal. Leaf size=194 \[ -\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{4 c d \left (c+d x^2\right )^2}-\frac {(b c-a d) (4 b c+3 a d) x \sqrt {a+b x^2}}{8 c^2 d^2 \left (c+d x^2\right )}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {\sqrt {b c-a d} \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} d^3} \]

[Out]

-1/4*(-a*d+b*c)*x*(b*x^2+a)^(3/2)/c/d/(d*x^2+c)^2+b^(5/2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/d^3-1/8*(3*a^2*d^
2+4*a*b*c*d+8*b^2*c^2)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))*(-a*d+b*c)^(1/2)/c^(5/2)/d^3-1/8*(-
a*d+b*c)*(3*a*d+4*b*c)*x*(b*x^2+a)^(1/2)/c^2/d^2/(d*x^2+c)

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Rubi [A]
time = 0.13, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {424, 540, 537, 223, 212, 385, 214} \begin {gather*} -\frac {\sqrt {b c-a d} \left (3 a^2 d^2+4 a b c d+8 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} d^3}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}-\frac {x \sqrt {a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((b*c - a*d)*x*(a + b*x^2)^(3/2))/(c*d*(c + d*x^2)^2) - ((b*c - a*d)*(4*b*c + 3*a*d)*x*Sqrt[a + b*x^2])/(
8*c^2*d^2*(c + d*x^2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^3 - (Sqrt[b*c - a*d]*(8*b^2*c^2 + 4*
a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*d^3)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 540

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/(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 - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))
*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]

Rubi steps

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

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Mathematica [A]
time = 1.48, size = 266, normalized size = 1.37 \begin {gather*} \frac {\frac {d x \sqrt {a+b x^2} \left (-a b c d \left (c-3 d x^2\right )-2 b^2 c^2 \left (2 c+3 d x^2\right )+a^2 d^2 \left (5 c+3 d x^2\right )\right )}{c^2 \left (c+d x^2\right )^2}-\frac {3 (-4 b c+a d)^2 \sqrt {-b c+a d} \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 )}{c^{5/2}}+\frac {4 b (10 b c-7 a d) \sqrt {b c-a d} \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 )}{c^{3/2}}-8 b^{5/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*x*Sqrt[a + b*x^2]*(-(a*b*c*d*(c - 3*d*x^2)) - 2*b^2*c^2*(2*c + 3*d*x^2) + a^2*d^2*(5*c + 3*d*x^2)))/(c^2*(
c + d*x^2)^2) - (3*(-4*b*c + a*d)^2*Sqrt[-(b*c) + a*d]*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(
Sqrt[c]*Sqrt[-(b*c) + a*d])])/c^(5/2) + (4*b*(10*b*c - 7*a*d)*Sqrt[b*c - a*d]*ArcTanh[(-(d*x*Sqrt[a + b*x^2])
+ Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[b*c - a*d])])/c^(3/2) - 8*b^(5/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8
*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10682\) vs. \(2(168)=336\).
time = 0.07, size = 10683, normalized size = 55.07

method	result	size
default	\(\text {Expression too large to display}\)	\(10683\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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. 352 vs. \(2 (168) = 336\).
time = 0.81, size = 1517, normalized size = 7.82 \begin {gather*} \left [\frac {16 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (8 \, b^{2} c^{4} + 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} + 4 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{c}} \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 (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (4 \, b^{2} c^{3} d + a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}}, -\frac {32 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{2} c^{4} + 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} + 4 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{c}} \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 (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (4 \, b^{2} c^{3} d + a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}}, \frac {{\left (8 \, b^{2} c^{4} + 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} + 4 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) + 8 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (4 \, b^{2} c^{3} d + a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}}, -\frac {16 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{2} c^{4} + 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} + 4 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) + 2 \, {\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (4 \, b^{2} c^{3} d + a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\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)^3,x, algorithm="fricas")

[Out]

[1/32*(16*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a
) + (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d
+ 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt((b*c - a*d)/c)*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*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2
*x^4 + 2*c*d*x^2 + c^2)) - 4*(3*(2*b^2*c^2*d^2 - a*b*c*d^3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2
*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2 + c^4*d^3), -1/32*(32*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*
x^2 + b^2*c^4)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2
*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt((b*c - a*d)/
c)*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*(a*c^2*x + (2*b*c^
2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*(3*(2*b^2*c^2*d^2 - a*b*
c*d^3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*
x^2 + c^4*d^3), 1/16*((8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4
+ 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(-(b*c - a*d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c
)*sqrt(b*x^2 + a)*sqrt(-(b*c - a*d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) + 8*(b^2*c^2*d^2*x^4 + 2*b^2
*c^3*d*x^2 + b^2*c^4)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(3*(2*b^2*c^2*d^2 - a*b*c*d^
3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2
+ c^4*d^3), -1/16*(16*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)
) - (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d
+ 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(-(b*c - a*d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)*
sqrt(-(b*c - a*d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) + 2*(3*(2*b^2*c^2*d^2 - a*b*c*d^3 - a^2*d^4)*x
^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2 + c^4*d^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (168) = 336\).
time = 0.52, size = 659, normalized size = 3.40 \begin {gather*} -\frac {b^{\frac {5}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, d^{3}} + \frac {{\left (8 \, b^{\frac {7}{2}} c^{3} - 4 \, a b^{\frac {5}{2}} c^{2} d - a^{2} b^{\frac {3}{2}} c d^{2} - 3 \, a^{3} \sqrt {b} d^{3}\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 \, \sqrt {-b^{2} c^{2} + a b c d} c^{2} d^{3}} - \frac {16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {7}{2}} c^{3} d - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {5}{2}} c^{2} d^{2} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {3}{2}} c d^{3} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} \sqrt {b} d^{4} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {9}{2}} c^{4} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {7}{2}} c^{3} d + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} c^{2} d^{2} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {3}{2}} c d^{3} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} \sqrt {b} d^{4} + 32 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {7}{2}} c^{3} d - 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} c^{2} d^{2} - 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {3}{2}} c d^{3} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} \sqrt {b} d^{4} + 6 \, a^{4} b^{\frac {5}{2}} c^{2} d^{2} - 3 \, a^{5} b^{\frac {3}{2}} c d^{3} - 3 \, a^{6} \sqrt {b} d^{4}}{4 \, {\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} c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^(5/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/d^3 + 1/8*(8*b^(7/2)*c^3 - 4*a*b^(5/2)*c^2*d - a^2*b^(3/2)*c
*d^2 - 3*a^3*sqrt(b)*d^3)*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
))/(sqrt(-b^2*c^2 + a*b*c*d)*c^2*d^3) - 1/4*(16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(7/2)*c^3*d - 20*(sqrt(b)*x
- sqrt(b*x^2 + a))^6*a*b^(5/2)*c^2*d^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(3/2)*c*d^3 + 3*(sqrt(b)*x - sq
rt(b*x^2 + a))^6*a^3*sqrt(b)*d^4 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(9/2)*c^4 - 72*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*a*b^(7/2)*c^3*d + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*c^2*d^2 + 15*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*a^3*b^(3/2)*c*d^3 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*sqrt(b)*d^4 + 32*(sqrt(b)*x - sqrt(b*x^2 +
a))^2*a^2*b^(7/2)*c^3*d - 28*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(5/2)*c^2*d^2 - 13*(sqrt(b)*x - sqrt(b*x^2
+ a))^2*a^4*b^(3/2)*c*d^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*sqrt(b)*d^4 + 6*a^4*b^(5/2)*c^2*d^2 - 3*a^5*
b^(3/2)*c*d^3 - 3*a^6*sqrt(b)*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*c^2*d^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\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((a + b*x^2)^(5/2)/(c + d*x^2)^3,x)

[Out]

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

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