3.58 \(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=131 \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^2}-\frac {\sqrt {b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^2}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {413, 523, 217, 206, 377, 208} \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^2}-\frac {\sqrt {b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^2}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c - a*d)*x*Sqrt[a + b*x^2])/(2*c*d*(c + d*x^2)) + (b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^2 - (S
qrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*d^2)

Rule 206

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

Rule 208

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

Rule 217

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 377

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 413

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 523

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]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 142, normalized size = 1.08 \[ \frac {\frac {\left (a^2 d^2+a b c d-2 b^2 c^2\right ) \tan ^{-1}\left (\frac {x \sqrt {a d-b c}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} \sqrt {a d-b c}}+2 b^{3/2} \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )-\frac {d x \sqrt {a+b x^2} (b c-a d)}{c \left (c+d x^2\right )}}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.97, size = 907, normalized size = 6.92 \[ \left [-\frac {4 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x - 4 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\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 )}{8 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {4 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x + 8 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\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 )}{8 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\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 (b c d x^{2} + b c^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x + 4 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\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 )}{4 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.68, size = 317, normalized size = 2.42 \[ -\frac {b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, d^{2}} + \frac {{\left (2 \, b^{\frac {5}{2}} c^{2} - a b^{\frac {3}{2}} c d - a^{2} \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, \sqrt {-b^{2} c^{2} + a b c d} c d^{2}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{2} + a^{2} b^{\frac {3}{2}} c d - a^{3} \sqrt {b} d^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 4621, normalized size = 35.27 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8/c*b/(a*d-b*c)*a*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x+1/2/(-
c*d)^(1/2)/d/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)
*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*a*b-1/4
/(-c*d)^(1/2)*c/d^2/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d
)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*
b^2-1/4/c/d*(-c*d)^(1/2)*b/(a*d-b*c)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d
)^(3/2)-3/8/c*b/(a*d-b*c)*a*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x
-1/2/(-c*d)^(1/2)/d/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/
d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))
*a*b+1/4/(-c*d)^(1/2)*c/d^2/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a
*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(
1/2)/d))*b^2+1/4/c/d*(-c*d)^(1/2)*b/(a*d-b*c)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a
*d-b*c)/d)^(3/2)+1/4/c/(a*d-b*c)/(x+(-c*d)^(1/2)/d)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*
b/d+(a*d-b*c)/d)^(5/2)-1/4/(-c*d)^(1/2)/c*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b
*c)/d)^(1/2)*a+1/4/(-c*d)^(1/2)/d*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(
1/2)*b+1/4/(-c*d)^(1/2)/c*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*a-1
/4/(-c*d)^(1/2)/d*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*b+1/4/c/(a*
d-b*c)/(x-(-c*d)^(1/2)/d)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(5/2)-3/4
/c/d*(-c*d)^(1/2)*b/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*(
(a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)
^(1/2)/d))*a^2+3/4/c/d*(-c*d)^(1/2)*b/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+
2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)
^(1/2))/(x-(-c*d)^(1/2)/d))*a^2+3/8/d*b^2/(a*d-b*c)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*
b/d+(a*d-b*c)/d)^(1/2)*x+9/8/d*b^(3/2)/(a*d-b*c)*ln(((x-(-c*d)^(1/2)/d)*b+(-c*d)^(1/2)*b/d)/b^(1/2)+((x-(-c*d)
^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))*a-1/4/d^2*b^(3/2)*ln(((x+(-c*d)^(1/2)/
d)*b-(-c*d)^(1/2)*b/d)/b^(1/2)+((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2
))+1/12/(-c*d)^(1/2)/c*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)-1/4/d^
2*b^(3/2)*ln(((x-(-c*d)^(1/2)/d)*b+(-c*d)^(1/2)*b/d)/b^(1/2)+((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^
(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))-1/12/(-c*d)^(1/2)/c*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)
*b/d+(a*d-b*c)/d)^(3/2)-3/4*c/d^2*b^(5/2)/(a*d-b*c)*ln(((x+(-c*d)^(1/2)/d)*b-(-c*d)^(1/2)*b/d)/b^(1/2)+((x+(-c
*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))-1/4/c*b/(a*d-b*c)*((x+(-c*d)^(1/2)/
d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x-3/8/c*b^(1/2)/(a*d-b*c)*a^2*ln(((x+(-c*d)^(1
/2)/d)*b-(-c*d)^(1/2)*b/d)/b^(1/2)+((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^
(1/2))+3/8/d*b^2/(a*d-b*c)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x-
1/4/c*b/(a*d-b*c)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(3/2)*x-3/8/c*b^(
1/2)/(a*d-b*c)*a^2*ln(((x-(-c*d)^(1/2)/d)*b+(-c*d)^(1/2)*b/d)/b^(1/2)+((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(
x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))+1/8/c*b/d*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*
b/d+(a*d-b*c)/d)^(1/2)*x+3/8/c/d*b^(1/2)*ln(((x+(-c*d)^(1/2)/d)*b-(-c*d)^(1/2)*b/d)/b^(1/2)+((x+(-c*d)^(1/2)/d
)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))*a+1/4/(-c*d)^(1/2)/c/((a*d-b*c)/d)^(1/2)*ln((-
2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/
2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*a^2+1/8/c*b/d*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*
d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*x+3/8/c/d*b^(1/2)*ln(((x-(-c*d)^(1/2)/d)*b+(-c*d)^(1/2)*b/d
)/b^(1/2)+((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))*a-1/4/(-c*d)^(1/2)
/c/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d
)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*a^2+9/8/d*b^(3/2)
/(a*d-b*c)*ln(((x+(-c*d)^(1/2)/d)*b-(-c*d)^(1/2)*b/d)/b^(1/2)+((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)
^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))*a-3/4/d^2*(-c*d)^(1/2)*b^2/(a*d-b*c)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*
(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)+3/4/d^2*(-c*d)^(1/2)*b^2/(a*d-b*c)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^
(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)-3/4*c/d^2*b^(5/2)/(a*d-b*c)*ln(((x-(-c*d)^(1/2)/d)*b+(-c*d)^(1
/2)*b/d)/b^(1/2)+((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))-3/4/c/d*(-c
*d)^(1/2)*b/(a*d-b*c)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*a-3/2/d
^2*(-c*d)^(1/2)*b^2/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((
a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^
(1/2)/d))*a+3/4*c/d^3*(-c*d)^(1/2)*b^3/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d
+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*(-c*d)^(1/2)*(x-(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d
)^(1/2))/(x-(-c*d)^(1/2)/d))+3/4/c/d*(-c*d)^(1/2)*b/(a*d-b*c)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)
^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2)*a+3/2/d^2*(-c*d)^(1/2)*b^2/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((-2*(-c*d)^(1/2)*
(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d)^(1/2)*(x+(-c*d)^(1
/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*a-3/4*c/d^3*(-c*d)^(1/2)*b^3/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*
ln((-2*(-c*d)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+2*(a*d-b*c)/d+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*(-c*d
)^(1/2)*(x+(-c*d)^(1/2)/d)*b/d+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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