\(\int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx\) [572]

Optimal result

Integrand size = 21, antiderivative size = 154 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx=-\frac {2 c \sqrt {c+d x}}{b}-\frac {2 (c+d x)^{3/2}}{3 b}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{b^{7/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{b^{7/4}} \] Output:

-2*c*(d*x+c)^(1/2)/b-2/3*(d*x+c)^(3/2)/b+(b^(1/2)*c-a^(1/2)*d)^(3/2)*arcta 
nh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(7/4)+(b^(1/2)*c+a 
^(1/2)*d)^(3/2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2)) 
/b^(7/4)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.38 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx=\frac {-2 b \sqrt {c+d x} (4 c+d x)-3 \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )+\frac {3 \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{3 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {561, 25, 27, 1602, 27, 25, 1602, 25, 25, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1602

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1602

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {-\frac {2 \left (d x +4 c \right ) \sqrt {d x +c}}{3}+\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{b}\)	\(197\)
risch	\(-\frac {2 \left (d x +4 c \right ) \sqrt {d x +c}}{3 b}+\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\)	\(202\)
derivativedivides	\(-\frac {2 \left (\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+c \sqrt {d x +c}\right )}{b}-\frac {\left (-2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (2 a b c \,d^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\)	\(207\)
default	\(-\frac {2 \left (\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+c \sqrt {d x +c}\right )}{b}+\frac {\left (2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\)	\(211\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (112) = 224\).

Time = 0.10 (sec) , antiderivative size = 926, normalized size of antiderivative = 6.01 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(3*b*sqrt((b*c^3 + 3*a*c*d^2 + b^3*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^ 
2*d^4 + a^3*d^6)/b^7))/b^3)*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqr 
t(d*x + c) + (3*b^3*c^3 + a*b^2*c*d^2 - b^5*sqrt((9*a*b^2*c^4*d^2 + 6*a^2* 
b*c^2*d^4 + a^3*d^6)/b^7))*sqrt((b*c^3 + 3*a*c*d^2 + b^3*sqrt((9*a*b^2*c^4 
*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))/b^3)) - 3*b*sqrt((b*c^3 + 3*a*c*d^ 
2 + b^3*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))/b^3)*log( 
-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) - (3*b^3*c^3 + a*b^2* 
c*d^2 - b^5*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))*sqrt( 
(b*c^3 + 3*a*c*d^2 + b^3*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6 
)/b^7))/b^3)) + 3*b*sqrt((b*c^3 + 3*a*c*d^2 - b^3*sqrt((9*a*b^2*c^4*d^2 + 
6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))/b^3)*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^ 
2*d^4)*sqrt(d*x + c) + (3*b^3*c^3 + a*b^2*c*d^2 + b^5*sqrt((9*a*b^2*c^4*d^ 
2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))*sqrt((b*c^3 + 3*a*c*d^2 - b^3*sqrt((9 
*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7))/b^3)) - 3*b*sqrt((b*c^3 
+ 3*a*c*d^2 - b^3*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b^7)) 
/b^3)*log(-(3*b^2*c^4 - 2*a*b*c^2*d^2 - a^2*d^4)*sqrt(d*x + c) - (3*b^3*c^ 
3 + a*b^2*c*d^2 + b^5*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 + a^3*d^6)/b 
^7))*sqrt((b*c^3 + 3*a*c*d^2 - b^3*sqrt((9*a*b^2*c^4*d^2 + 6*a^2*b*c^2*d^4 
 + a^3*d^6)/b^7))/b^3)) + 4*(d*x + 4*c)*sqrt(d*x + c))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (112) = 224\).

Time = 0.16 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.36 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx=-\frac {{\left (2 \, \sqrt {a b} b^{3} c^{2} d^{2} - {\left (\sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} b^{2} d^{2} + {\left (b^{3} c^{3} - a b^{2} c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{4} c + \sqrt {b^{8} c^{2} - {\left (b^{4} c^{2} - a b^{3} d^{2}\right )} b^{4}}}{b^{4}}}}\right )}{{\left (b^{4} c - \sqrt {a b} b^{3} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (2 \, \sqrt {a b} b^{3} c^{2} d^{2} - {\left (\sqrt {a b} b c^{2} + \sqrt {a b} a d^{2}\right )} b^{2} d^{2} - {\left (b^{3} c^{3} - a b^{2} c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{4} c - \sqrt {b^{8} c^{2} - {\left (b^{4} c^{2} - a b^{3} d^{2}\right )} b^{4}}}{b^{4}}}}\right )}{{\left (b^{4} c + \sqrt {a b} b^{3} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} - \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{2} + 3 \, \sqrt {d x + c} b^{2} c\right )}}{3 \, b^{3}} \] Input:

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

Output:

-(2*sqrt(a*b)*b^3*c^2*d^2 - (sqrt(a*b)*b*c^2 + sqrt(a*b)*a*d^2)*b^2*d^2 + 
(b^3*c^3 - a*b^2*c*d^2)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(b^4*c + 
 sqrt(b^8*c^2 - (b^4*c^2 - a*b^3*d^2)*b^4))/b^4))/((b^4*c - sqrt(a*b)*b^3* 
d)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(d)) + (2*sqrt(a*b)*b^3*c^2*d^2 - (sqrt 
(a*b)*b*c^2 + sqrt(a*b)*a*d^2)*b^2*d^2 - (b^3*c^3 - a*b^2*c*d^2)*abs(b)*ab 
s(d))*arctan(sqrt(d*x + c)/sqrt(-(b^4*c - sqrt(b^8*c^2 - (b^4*c^2 - a*b^3* 
d^2)*b^4))/b^4))/((b^4*c + sqrt(a*b)*b^3*d)*sqrt(-b^2*c + sqrt(a*b)*b*d)*a 
bs(d)) - 2/3*((d*x + c)^(3/2)*b^2 + 3*sqrt(d*x + c)*b^2*c)/b^3
 

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1511, normalized size of antiderivative = 9.81 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((a^3*d^6*(c + d*x)^(1/2)*(c^3/(4*b^2) + (a*d^3*(a*b^7)^(1/2))/(4*b^ 
7) + (3*c^2*d*(a*b^7)^(1/2))/(4*b^6) + (3*a*c*d^2)/(4*b^3))^(1/2)*32i)/((1 
6*a^4*d^8)/b^2 - 48*a^2*c^4*d^4 + (32*a^3*c^2*d^6)/b + (32*a^2*c^3*d^5*(a* 
b^7)^(1/2))/b^4 - (48*a*c^5*d^3*(a*b^7)^(1/2))/b^3 + (16*a^3*c*d^7*(a*b^7) 
^(1/2))/b^5) - (a*c^3*d^3*(a*b^7)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*b^2) + (a* 
d^3*(a*b^7)^(1/2))/(4*b^7) + (3*c^2*d*(a*b^7)^(1/2))/(4*b^6) + (3*a*c*d^2) 
/(4*b^3))^(1/2)*96i)/(16*a^4*d^8 + 32*a^3*b*c^2*d^6 - 48*a^2*b^2*c^4*d^4 + 
 (32*a^2*c^3*d^5*(a*b^7)^(1/2))/b^2 - (48*a*c^5*d^3*(a*b^7)^(1/2))/b + (16 
*a^3*c*d^7*(a*b^7)^(1/2))/b^3) - (a^2*c*d^5*(a*b^7)^(1/2)*(c + d*x)^(1/2)* 
(c^3/(4*b^2) + (a*d^3*(a*b^7)^(1/2))/(4*b^7) + (3*c^2*d*(a*b^7)^(1/2))/(4* 
b^6) + (3*a*c*d^2)/(4*b^3))^(1/2)*32i)/(16*a^4*b*d^8 - 48*a*c^5*d^3*(a*b^7 
)^(1/2) - 48*a^2*b^3*c^4*d^4 + 32*a^3*b^2*c^2*d^6 + (32*a^2*c^3*d^5*(a*b^7 
)^(1/2))/b + (16*a^3*c*d^7*(a*b^7)^(1/2))/b^2) + (a^2*b*c^2*d^4*(c + d*x)^ 
(1/2)*(c^3/(4*b^2) + (a*d^3*(a*b^7)^(1/2))/(4*b^7) + (3*c^2*d*(a*b^7)^(1/2 
))/(4*b^6) + (3*a*c*d^2)/(4*b^3))^(1/2)*96i)/((16*a^4*d^8)/b^2 - 48*a^2*c^ 
4*d^4 + (32*a^3*c^2*d^6)/b + (32*a^2*c^3*d^5*(a*b^7)^(1/2))/b^4 - (48*a*c^ 
5*d^3*(a*b^7)^(1/2))/b^3 + (16*a^3*c*d^7*(a*b^7)^(1/2))/b^5))*((b^5*c^3 + 
a*d^3*(a*b^7)^(1/2) + 3*a*b^4*c*d^2 + 3*b*c^2*d*(a*b^7)^(1/2))/(4*b^7))^(1 
/2)*2i - atan((a^3*d^6*(c + d*x)^(1/2)*(c^3/(4*b^2) - (a*d^3*(a*b^7)^(1/2) 
)/(4*b^7) - (3*c^2*d*(a*b^7)^(1/2))/(4*b^6) + (3*a*c*d^2)/(4*b^3))^(1/2...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.76 \[ \int \frac {x (c+d x)^{3/2}}{a-b x^2} \, dx=\frac {-6 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) d +6 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c -3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d +3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) d -3 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c +3 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c -16 \sqrt {d x +c}\, b c -4 \sqrt {d x +c}\, b d x}{6 b^{2}} \] Input:

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

Output:

( - 6*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)))*d + 6*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)))*c - 3* 
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))*d + 3*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))*d - 3*sqrt(b)*sqrt( 
sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sq 
rt(c + d*x))*c + 3*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)* 
sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*c - 16*sqrt(c + d*x)*b*c - 4*sqr 
t(c + d*x)*b*d*x)/(6*b**2)