\(\int \frac {x^2 (c+d x^2)^{3/2}}{a+b x^2} \, dx\) [687]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\sqrt {a} (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {488, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}}-\frac {\sqrt {a} (b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {x \sqrt {c+d x^2} (5 b c-4 a d)}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b} \]

[In]

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

[Out]

((5*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^2) + (d*x^3*Sqrt[c + d*x^2])/(4*b) - (Sqrt[a]*(b*c - a*d)^(3/2)*ArcTa
n[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^3 + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]
*x)/Sqrt[c + d*x^2]])/(8*b^3*Sqrt[d])

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 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 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 596

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

Rubi steps \begin{align*} \text {integral}& = \frac {d x^3 \sqrt {c+d x^2}}{4 b}+\frac {\int \frac {x^2 \left (c (4 b c-3 a d)+d (5 b c-4 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 b} \\ & = \frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\int \frac {a c d (5 b c-4 a d)-d \left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 b^2 d} \\ & = \frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\left (a (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^3} \\ & = \frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\left (a (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^3} \\ & = \frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\sqrt {a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(158)=316\).

Time = 1.64 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.54 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {b x \sqrt {c+d x^2} \left (5 b c-4 a d+2 b d x^2\right )+\frac {8 (b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (b c-a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} d}+\frac {8 (-b c+a d) \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} d}+\frac {2 \left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{\sqrt {d}}}{8 b^3} \]

[In]

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

[Out]

(b*x*Sqrt[c + d*x^2]*(5*b*c - 4*a*d + 2*b*d*x^2) + (8*(b*c - a*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*
c - a*d]]*(b*c - a*d + Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d])*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c
- a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*d) + (8*(-(b*c) + a*d)*(-(b*c) + a*d + Sqrt[b]*Sq
rt[c]*Sqrt[b*c - a*d])*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt
[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*d) + (2*(3*b^2*c^2 - 12*a*b*
c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/Sqrt[d])/(8*b^3)

Maple [A] (verified)

Time = 2.98 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+4 a d -5 b c \right ) x}{4}-\frac {\left (8 a^{2} d^{2}-12 a b c d +3 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{4 \sqrt {d}}+\frac {2 a \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{3}}\)	\(133\)
risch	\(-\frac {x \left (-2 b d \,x^{2}+4 a d -5 b c \right ) \sqrt {d \,x^{2}+c}}{8 b^{2}}+\frac {\frac {\left (8 a^{2} d^{2}-12 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {4 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {4 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 b^{2}}\)	\(439\)
default	\(\text {Expression too large to display}\)	\(1306\)

[In]

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

[Out]

-1/2/b^3*(1/4*b*(d*x^2+c)^(1/2)*(-2*b*d*x^2+4*a*d-5*b*c)*x-1/4*(8*a^2*d^2-12*a*b*c*d+3*b^2*c^2)/d^(1/2)*arctan
h((d*x^2+c)^(1/2)/x/d^(1/2))+2*a*(a*d-b*c)^2/((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 = 0.57 (sec) , antiderivative size = 894, normalized size of antiderivative = 5.66 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\left [\frac {{\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 4 \, \sqrt {-a b c + a^{2} d} {\left (b c d - a d^{2}\right )} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d}, -\frac {{\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + 2 \, \sqrt {-a b c + a^{2} d} {\left (b c d - a d^{2}\right )} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d}, -\frac {8 \, \sqrt {a b c - a^{2} d} {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - {\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d}, -\frac {4 \, \sqrt {a b c - a^{2} d} {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + {\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d}\right ] \]

[In]

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

[Out]

[1/16*((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 4*sqrt(-
a*b*c + a^2*d)*(b*c*d - a*d^2)*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*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(
2*b^2*d^2*x^3 + (5*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 + c))/(b^3*d), -1/8*((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2
)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + 2*sqrt(-a*b*c + a^2*d)*(b*c*d - a*d^2)*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*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^
2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (2*b^2*d^2*x^3 + (5*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 +
c))/(b^3*d), -1/16*(8*sqrt(a*b*c - a^2*d)*(b*c*d - a*d^2)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 -
a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - (3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)
*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 2*(2*b^2*d^2*x^3 + (5*b^2*c*d - 4*a*b*d^2)*x)*sqrt(
d*x^2 + c))/(b^3*d), -1/8*(4*sqrt(a*b*c - a^2*d)*(b*c*d - a*d^2)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)
*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + (3*b^2*c^2 - 12*a*b*c*d + 8*a
^2*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (2*b^2*d^2*x^3 + (5*b^2*c*d - 4*a*b*d^2)*x)*sqrt(d*x^2 +
c))/(b^3*d)]

Sympy [F]

\[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}{a + b x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{b x^{2} + a} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {x^2\,{\left (d\,x^2+c\right )}^{3/2}}{b\,x^2+a} \,d x \]

[In]

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

[Out]

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