\(\int \frac {(c+d x^2)^{5/2}}{(a+b x^2)^2} \, dx\) [753]

Optimal result

Integrand size = 21, antiderivative size = 174 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {d (b c-2 a d) x \sqrt {c+d x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^{3/2} (b c+4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac {d^{3/2} (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {424, 542, 537, 223, 212, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-a d)^{3/2} (4 a d+b c) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac {d^{3/2} (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3}-\frac {d x \sqrt {c+d x^2} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[In]

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

[Out]

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

Rule 211

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

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 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 542

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

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {b x \sqrt {c+d x^2} \left (b^2 c^2+2 a^2 d^2+a b d \left (-2 c+d x^2\right )\right )}{a \left (a+b x^2\right )}-\frac {\sqrt {b c-a d} \left (b^2 c^2+3 a b c d-4 a^2 d^2\right ) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{3/2}}+d^{3/2} (-5 b c+4 a d) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{2 b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {-d^{\frac {3}{2}} \left (\sqrt {d \,x^{2}+c}\, b x \sqrt {d}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) a d +5 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) b c \right )+\frac {\left (a d -b c \right )^{2} \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d +b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{a}}{2 b^{3}}\)	\(156\)
risch	\(\text {Expression too large to display}\)	\(1029\)
default	\(\text {Expression too large to display}\)	\(5290\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.67 (sec) , antiderivative size = 1228, normalized size of antiderivative = 7.06 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (146) = 292\).

Time = 0.30 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.34 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} d^{2} x}{2 \, b^{2}} - \frac {{\left (5 \, b c d^{\frac {3}{2}} - 4 \, a d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3}} - \frac {{\left (b^{3} c^{3} \sqrt {d} + 2 \, a b^{2} c^{2} d^{\frac {3}{2}} - 7 \, a^{2} b c d^{\frac {5}{2}} + 4 \, a^{3} d^{\frac {7}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a b^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} - b^{3} c^{4} \sqrt {d} + 2 \, a b^{2} c^{3} d^{\frac {3}{2}} - a^{2} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a b^{3}} \]

[In]

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

[Out]

1/2*sqrt(d*x^2 + c)*d^2*x/b^2 - 1/4*(5*b*c*d^(3/2) - 4*a*d^(5/2))*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/b^3 - 1
/2*(b^3*c^3*sqrt(d) + 2*a*b^2*c^2*d^(3/2) - 7*a^2*b*c*d^(5/2) + 4*a^3*d^(7/2))*arctan(1/2*((sqrt(d)*x - sqrt(d
*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a*b^3) - ((sqrt(d)*x - sqrt(d*
x^2 + c))^2*b^3*c^3*sqrt(d) - 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b^2*c^2*d^(3/2) + 5*(sqrt(d)*x - sqrt(d*x^2
+ c))^2*a^2*b*c*d^(5/2) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*d^(7/2) - b^3*c^4*sqrt(d) + 2*a*b^2*c^3*d^(3/2
) - a^2*b*c^2*d^(5/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d
)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*a*b^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

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

[Out]

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