\(\int \frac {x^2 (c+d x)^{5/2}}{(a-b x^2)^3} \, dx\) [708]

Optimal result

Integrand size = 23, antiderivative size = 284 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\frac {\sqrt {c+d x} \left (b c^2 x+a d (2 c+d x)\right )}{4 b^2 \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (2 b c^2 x+a d (17 c+11 d x)\right )}{16 a b^2 \left (a-b x^2\right )}+\frac {\sqrt {\sqrt {b} c-\sqrt {a} d} \left (4 b c^2+2 \sqrt {a} \sqrt {b} c d-21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{3/2} b^{11/4}}-\frac {\sqrt {\sqrt {b} c+\sqrt {a} d} \left (4 b c^2-2 \sqrt {a} \sqrt {b} c d-21 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{3/2} b^{11/4}} \] Output:

1/4*(d*x+c)^(1/2)*(b*c^2*x+a*d*(d*x+2*c))/b^2/(-b*x^2+a)^2-1/16*(d*x+c)^(1 
/2)*(2*b*c^2*x+a*d*(11*d*x+17*c))/a/b^2/(-b*x^2+a)+1/32*(b^(1/2)*c-a^(1/2) 
*d)^(1/2)*(4*b*c^2+2*a^(1/2)*b^(1/2)*c*d-21*a*d^2)*arctanh(b^(1/4)*(d*x+c) 
^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(3/2)/b^(11/4)-1/32*(b^(1/2)*c+a^(1/ 
2)*d)^(1/2)*(4*b*c^2-2*a^(1/2)*b^(1/2)*c*d-21*a*d^2)*arctanh(b^(1/4)*(d*x+ 
c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(3/2)/b^(11/4)
 

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} b \sqrt {c+d x} \left (2 b^2 c^2 x^3-a^2 d (9 c+7 d x)+a b x \left (2 c^2+17 c d x+11 d^2 x^2\right )\right )}{\left (a-b x^2\right )^2}+\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (4 b c^2-2 \sqrt {a} \sqrt {b} c d-21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )-\sqrt {-b c+\sqrt {a} \sqrt {b} d} \left (4 b c^2+2 \sqrt {a} \sqrt {b} c d-21 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{32 a^{3/2} b^3} \] Input:

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

Output:

((2*Sqrt[a]*b*Sqrt[c + d*x]*(2*b^2*c^2*x^3 - a^2*d*(9*c + 7*d*x) + a*b*x*( 
2*c^2 + 17*c*d*x + 11*d^2*x^2)))/(a - b*x^2)^2 + Sqrt[-(b*c) - Sqrt[a]*Sqr 
t[b]*d]*(4*b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - 21*a*d^2)*ArcTan[(Sqrt[-(b*c) - 
 Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)] - Sqrt[-(b*c) 
+ Sqrt[a]*Sqrt[b]*d]*(4*b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - 21*a*d^2)*ArcTan[( 
Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/ 
(32*a^(3/2)*b^3)
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 27, 1672, 27, 2206, 27, 27, 1480, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 561

\(\displaystyle \frac {2 \int \frac {x^2 (c+d x)^3}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {d^2 x^2 (c+d x)^3}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d^3}\)

\(\Big \downarrow \) 1672

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {2 \left (8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-a \left (-\frac {5 b c^4}{d^2}+8 a c^2-\frac {3 a^2 d^2}{b}\right ) (c+d x)+\frac {a c \left (b c^2-a d^2\right )^2}{b d^2}\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \int \frac {8 a \left (a-\frac {b c^2}{d^2}\right ) (c+d x)^3-a \left (-\frac {5 b c^4}{d^2}+8 a c^2-\frac {3 a^2 d^2}{b}\right ) (c+d x)+\frac {a c \left (b c^2-a d^2\right )^2}{b d^2}}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {d^4 \int -\frac {2 a \left (2 c \left (b c^2-a d^2\right )^3+\left (2 b c^2-21 a d^2\right ) (c+d x) \left (b c^2-a d^2\right )^2\right )}{d^6 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (2 c \left (b c^2-3 a d^2\right )-(c+d x) \left (11 a d^2+2 b c^2\right )\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (2 c \left (b c^2-3 a d^2\right )-(c+d x) \left (11 a d^2+2 b c^2\right )\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\int \frac {\left (b c^2-a d^2\right )^2 \left (2 c \left (b c^2-a d^2\right )+\left (2 b c^2-21 a d^2\right ) (c+d x)\right )}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 b d^2 \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (2 c \left (b c^2-3 a d^2\right )-(c+d x) \left (11 a d^2+2 b c^2\right )\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \int \frac {2 c \left (b c^2-a d^2\right )+\left (2 b c^2-21 a d^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (2 c \left (b c^2-3 a d^2\right )-(c+d x) \left (11 a d^2+2 b c^2\right )\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \left (\frac {1}{2} \left (-\frac {4 b^{3/2} c^3}{\sqrt {a} d}+23 \sqrt {a} \sqrt {b} c d-21 a d^2+2 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}+\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (-2 \sqrt {a} \sqrt {b} c d-21 a d^2+4 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (\frac {d^4 \left (\frac {\sqrt {c+d x} \left (b c^2-a d^2\right ) \left (2 c \left (b c^2-3 a d^2\right )-(c+d x) \left (11 a d^2+2 b c^2\right )\right )}{4 b d^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {\left (b c^2-a d^2\right ) \left (\frac {d \sqrt {\sqrt {a} d+\sqrt {b} c} \left (-2 \sqrt {a} \sqrt {b} c d-21 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4}}+\frac {d^2 \left (-\frac {4 b^{3/2} c^3}{\sqrt {a} d}+23 \sqrt {a} \sqrt {b} c d-21 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b d^2}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (b c^2-a d^2\right )^2-(c+d x) \left (b^2 c^4-a^2 d^4\right )\right )}{8 b^2 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^3}\)

Input:

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

Output:

(2*(-1/8*(d^2*Sqrt[c + d*x]*(c*(b*c^2 - a*d^2)^2 - (b^2*c^4 - a^2*d^4)*(c 
+ d*x)))/(b^2*(b*c^2 - a*d^2)*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - ( 
b*(c + d*x)^2)/d^2)^2) + (d^4*(((b*c^2 - a*d^2)*Sqrt[c + d*x]*(2*c*(b*c^2 
- 3*a*d^2) - (2*b*c^2 + 11*a*d^2)*(c + d*x)))/(4*b*d^2*(a - (b*c^2)/d^2 + 
(2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - ((b*c^2 - a*d^2)*((d^2*(2* 
b*c^2 - (4*b^(3/2)*c^3)/(Sqrt[a]*d) + 23*Sqrt[a]*Sqrt[b]*c*d - 21*a*d^2)*A 
rcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(3/4)*Sq 
rt[Sqrt[b]*c - Sqrt[a]*d]) + (d*Sqrt[Sqrt[b]*c + Sqrt[a]*d]*(4*b*c^2 - 2*S 
qrt[a]*Sqrt[b]*c*d - 21*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b 
]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(3/4))))/(4*b*d^2)))/(8*a*b*(b*c^2 - a*d^2 
))))/d^3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1672

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q 
, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a 
*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), 
 x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)* 
Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + 
 c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(-\frac {9 \left (-\frac {23 d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (-21 a \,d^{2}+2 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{23}+b c \left (a \,d^{2}-\frac {4 b \,c^{2}}{23}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{18}+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\frac {23 d \left (\frac {\left (21 a \,d^{2}-2 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{23}+b c \left (a \,d^{2}-\frac {4 b \,c^{2}}{23}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{18}+\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \sqrt {a b \,d^{2}}\, \left (-\frac {2 b^{2} c^{2} x^{3}}{9}-\frac {2 x \left (\frac {11}{2} d^{2} x^{2}+\frac {17}{2} c d x +c^{2}\right ) a b}{9}+a^{2} d \left (c +\frac {7 d x}{9}\right )\right )\right )\right )}{16 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, a \,b^{2} \left (-b \,x^{2}+a \right )^{2}}\)	\(330\)
default	\(2 d^{3} \left (\frac {\frac {\left (11 a \,d^{2}+2 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 a b \,d^{2}}-\frac {c \left (8 a \,d^{2}+3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{16 b a \,d^{2}}-\frac {\left (7 a^{2} d^{4}-b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} a \,d^{2}}-\frac {\left (a \,d^{2}-b \,c^{2}\right )^{2} c \sqrt {d x +c}}{16 a \,b^{2} d^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\frac {\left (23 a b c \,d^{2}-4 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+2 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-23 a b c \,d^{2}+4 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+2 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 a b \,d^{2}}\right )\)	\(397\)
derivativedivides	\(-2 d^{3} \left (-\frac {\frac {\left (11 a \,d^{2}+2 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 a b \,d^{2}}-\frac {c \left (8 a \,d^{2}+3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {5}{2}}}{16 b a \,d^{2}}-\frac {\left (7 a^{2} d^{4}-b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b^{2} a \,d^{2}}-\frac {\left (a \,d^{2}-b \,c^{2}\right )^{2} c \sqrt {d x +c}}{16 a \,b^{2} d^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\frac {\left (23 a b c \,d^{2}-4 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+2 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-23 a b c \,d^{2}+4 c^{3} b^{2}-21 \sqrt {a b \,d^{2}}\, a \,d^{2}+2 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{32 a b \,d^{2}}\right )\)	\(398\)

Input:

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

Output:

-9/16/(a*b*d^2)^(1/2)*(-23/18*d*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(1/23*(-21 
*a*d^2+2*b*c^2)*(a*b*d^2)^(1/2)+b*c*(a*d^2-4/23*b*c^2))*(-b*x^2+a)^2*arcta 
n(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+((-b*c+(a*b*d^2)^(1/2) 
)*b)^(1/2)*(-23/18*d*(1/23*(21*a*d^2-2*b*c^2)*(a*b*d^2)^(1/2)+b*c*(a*d^2-4 
/23*b*c^2))*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b) 
^(1/2))+((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(d*x+c)^(1/2)*(a*b*d^2)^(1/2)*(-2/ 
9*b^2*c^2*x^3-2/9*x*(11/2*d^2*x^2+17/2*c*d*x+c^2)*a*b+a^2*d*(c+7/9*d*x)))) 
/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/a/b^2/(- 
b*x^2+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1466 vs. \(2 (226) = 452\).

Time = 0.17 (sec) , antiderivative size = 1466, normalized size of antiderivative = 5.16 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*((a*b^4*x^4 - 2*a^2*b^3*x^2 + a^3*b^2)*sqrt((a^3*b^5*sqrt((6400*b^2*c 
^4*d^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a^3*b^11)) + 16*b^2*c^5 - 1 
80*a*b*c^3*d^2 + 525*a^2*c*d^4)/(a^3*b^5))*log(-(1280*b^3*c^6*d^3 - 20816* 
a*b^2*c^4*d^5 + 111132*a^2*b*c^2*d^7 - 194481*a^3*d^9)*sqrt(d*x + c) + (16 
0*a^2*b^4*c^3*d^4 - 882*a^3*b^3*c*d^6 - (4*a^3*b^9*c^2 - 21*a^4*b^8*d^2)*s 
qrt((6400*b^2*c^4*d^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a^3*b^11)))* 
sqrt((a^3*b^5*sqrt((6400*b^2*c^4*d^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10 
)/(a^3*b^11)) + 16*b^2*c^5 - 180*a*b*c^3*d^2 + 525*a^2*c*d^4)/(a^3*b^5))) 
- (a*b^4*x^4 - 2*a^2*b^3*x^2 + a^3*b^2)*sqrt((a^3*b^5*sqrt((6400*b^2*c^4*d 
^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a^3*b^11)) + 16*b^2*c^5 - 180*a 
*b*c^3*d^2 + 525*a^2*c*d^4)/(a^3*b^5))*log(-(1280*b^3*c^6*d^3 - 20816*a*b^ 
2*c^4*d^5 + 111132*a^2*b*c^2*d^7 - 194481*a^3*d^9)*sqrt(d*x + c) - (160*a^ 
2*b^4*c^3*d^4 - 882*a^3*b^3*c*d^6 - (4*a^3*b^9*c^2 - 21*a^4*b^8*d^2)*sqrt( 
(6400*b^2*c^4*d^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a^3*b^11)))*sqrt 
((a^3*b^5*sqrt((6400*b^2*c^4*d^6 - 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a 
^3*b^11)) + 16*b^2*c^5 - 180*a*b*c^3*d^2 + 525*a^2*c*d^4)/(a^3*b^5))) + (a 
*b^4*x^4 - 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(-(a^3*b^5*sqrt((6400*b^2*c^4*d^6 
- 70560*a*b*c^2*d^8 + 194481*a^2*d^10)/(a^3*b^11)) - 16*b^2*c^5 + 180*a*b* 
c^3*d^2 - 525*a^2*c*d^4)/(a^3*b^5))*log(-(1280*b^3*c^6*d^3 - 20816*a*b^2*c 
^4*d^5 + 111132*a^2*b*c^2*d^7 - 194481*a^3*d^9)*sqrt(d*x + c) + (160*a^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\int { -\frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{2}}{{\left (b x^{2} - a\right )}^{3}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (226) = 452\).

Time = 0.27 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.10 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\frac {{\left ({\left (2 \, b c^{2} d - 21 \, a d^{3}\right )} a^{2} d^{2} {\left | b \right |} + 2 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (4 \, a b^{2} c^{4} d - 23 \, a^{2} b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{3} c + \sqrt {a^{2} b^{6} c^{2} - {\left (a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} a b^{3}}}{a b^{3}}}}\right )}{32 \, {\left (a^{2} b^{3} d - \sqrt {a b} a b^{3} c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left ({\left (2 \, b c^{2} d - 21 \, a d^{3}\right )} a^{2} d^{2} {\left | b \right |} - 2 \, {\left (\sqrt {a b} b c^{3} d - \sqrt {a b} a c d^{3}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |} - {\left (4 \, a b^{2} c^{4} d - 23 \, a^{2} b c^{2} d^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b^{3} c - \sqrt {a^{2} b^{6} c^{2} - {\left (a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} a b^{3}}}{a b^{3}}}}\right )}{32 \, {\left (a^{2} b^{3} d + \sqrt {a b} a b^{3} c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {2 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c^{2} d - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{3} d + 6 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{4} d - 2 \, \sqrt {d x + c} b^{2} c^{5} d + 11 \, {\left (d x + c\right )}^{\frac {7}{2}} a b d^{3} - 16 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{3} + {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} + 4 \, \sqrt {d x + c} a b c^{3} d^{3} - 7 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{5} - 2 \, \sqrt {d x + c} a^{2} c d^{5}}{16 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )}^{2} a b^{2}} \] Input:

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

Output:

1/32*((2*b*c^2*d - 21*a*d^3)*a^2*d^2*abs(b) + 2*(sqrt(a*b)*b*c^3*d - sqrt( 
a*b)*a*c*d^3)*abs(a)*abs(b)*abs(d) - (4*a*b^2*c^4*d - 23*a^2*b*c^2*d^3)*ab 
s(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c + sqrt(a^2*b^6*c^2 - (a*b^3*c^2 
- a^2*b^2*d^2)*a*b^3))/(a*b^3)))/((a^2*b^3*d - sqrt(a*b)*a*b^3*c)*sqrt(-b^ 
2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) + 1/32*((2*b*c^2*d - 21*a*d^3)*a^2*d^2 
*abs(b) - 2*(sqrt(a*b)*b*c^3*d - sqrt(a*b)*a*c*d^3)*abs(a)*abs(b)*abs(d) - 
 (4*a*b^2*c^4*d - 23*a^2*b*c^2*d^3)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a* 
b^3*c - sqrt(a^2*b^6*c^2 - (a*b^3*c^2 - a^2*b^2*d^2)*a*b^3))/(a*b^3)))/((a 
^2*b^3*d + sqrt(a*b)*a*b^3*c)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(a)*abs(d)) 
+ 1/16*(2*(d*x + c)^(7/2)*b^2*c^2*d - 6*(d*x + c)^(5/2)*b^2*c^3*d + 6*(d*x 
 + c)^(3/2)*b^2*c^4*d - 2*sqrt(d*x + c)*b^2*c^5*d + 11*(d*x + c)^(7/2)*a*b 
*d^3 - 16*(d*x + c)^(5/2)*a*b*c*d^3 + (d*x + c)^(3/2)*a*b*c^2*d^3 + 4*sqrt 
(d*x + c)*a*b*c^3*d^3 - 7*(d*x + c)^(3/2)*a^2*d^5 - 2*sqrt(d*x + c)*a^2*c* 
d^5)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 - a*d^2)^2*a*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 2169, normalized size of antiderivative = 7.64 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(((11*a*d^3 + 2*b*c^2*d)*(c + d*x)^(7/2))/(16*a*b) + ((c + d*x)^(3/2)*(6*b 
^2*c^4*d - 7*a^2*d^5 + a*b*c^2*d^3))/(16*a*b^2) - ((c + d*x)^(1/2)*(a^2*c* 
d^5 + b^2*c^5*d - 2*a*b*c^3*d^3))/(8*a*b^2) - (c*(8*a*d^3 + 3*b*c^2*d)*(c 
+ d*x)^(5/2))/(8*a*b))/(b^2*(c + d*x)^4 + a^2*d^4 + b^2*c^4 + (6*b^2*c^2 - 
 2*a*b*d^2)*(c + d*x)^2 - (4*b^2*c^3 - 4*a*b*c*d^2)*(c + d*x) - 4*b^2*c*(c 
 + d*x)^3 - 2*a*b*c^2*d^2) - 2*atanh((441*a*d^8*(c + d*x)^(1/2)*(c^5/(256* 
a^3*b^3) + (525*c*d^4)/(4096*a*b^5) - (45*c^3*d^2)/(1024*a^2*b^4) - (441*d 
^5*(a^9*b^11)^(1/2))/(4096*a^5*b^11) + (5*c^2*d^3*(a^9*b^11)^(1/2))/(256*a 
^6*b^10))^(1/2))/(32*((5*c^6*d^5)/(32*a^2) - (9261*a*d^11)/(2048*b^3) + (1 
2705*c^2*d^9)/(2048*b^2) - (941*c^4*d^7)/(512*a*b) + (441*c*d^10*(a^9*b^11 
)^(1/2))/(1024*a^4*b^8) - (521*c^3*d^8*(a^9*b^11)^(1/2))/(1024*a^5*b^7) + 
(5*c^5*d^6*(a^9*b^11)^(1/2))/(64*a^6*b^6))) - (5*c^2*d^6*(c + d*x)^(1/2)*( 
c^5/(256*a^3*b^3) + (525*c*d^4)/(4096*a*b^5) - (45*c^3*d^2)/(1024*a^2*b^4) 
 - (441*d^5*(a^9*b^11)^(1/2))/(4096*a^5*b^11) + (5*c^2*d^3*(a^9*b^11)^(1/2 
))/(256*a^6*b^10))^(1/2))/(2*((12705*c^2*d^9)/(2048*b^3) - (9261*a*d^11)/( 
2048*b^4) - (941*c^4*d^7)/(512*a*b^2) + (5*c^6*d^5)/(32*a^2*b) + (441*c*d^ 
10*(a^9*b^11)^(1/2))/(1024*a^4*b^9) - (521*c^3*d^8*(a^9*b^11)^(1/2))/(1024 
*a^5*b^8) + (5*c^5*d^6*(a^9*b^11)^(1/2))/(64*a^6*b^7))) - (5*c^3*d^5*(a^9* 
b^11)^(1/2)*(c + d*x)^(1/2)*(c^5/(256*a^3*b^3) + (525*c*d^4)/(4096*a*b^5) 
- (45*c^3*d^2)/(1024*a^2*b^4) - (441*d^5*(a^9*b^11)^(1/2))/(4096*a^5*b^...
 

Reduce [B] (verification not implemented)

Time = 3.19 (sec) , antiderivative size = 1430, normalized size of antiderivative = 5.04 \[ \int \frac {x^2 (c+d x)^{5/2}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 42*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( 
b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*d**2 + 8*sqrt(a)*sqrt(sqrt(b)*sqrt 
(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c) 
))*a**2*b*c**2 + 84*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d 
*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*d**2*x**2 - 16*sqrt 
(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqr 
t(b)*sqrt(a)*d - b*c)))*a*b**2*c**2*x**2 - 42*sqrt(a)*sqrt(sqrt(b)*sqrt(a) 
*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))* 
a*b**2*d**2*x**4 + 8*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + 
d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c**2*x**4 + 4*sqrt(b 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt( 
b)*sqrt(a)*d - b*c)))*a**3*c*d - 8*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*c*d* 
x**2 + 4*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*c*d*x**4 - 21*sqrt(a)*sqrt(sqr 
t(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt( 
c + d*x))*a**3*d**2 + 4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt( 
sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2*b*c**2 + 42*sqrt(a) 
*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt 
(b)*sqrt(c + d*x))*a**2*b*d**2*x**2 - 8*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d ...