\(\int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx\) [580]

Optimal result

Integrand size = 20, antiderivative size = 167 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=-\frac {4 c d \sqrt {c+d x}}{b}-\frac {2 d (c+d x)^{3/2}}{3 b}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{\sqrt {a} b^{7/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{\sqrt {a} b^{7/4}} \] Output:

-4*c*d*(d*x+c)^(1/2)/b-2/3*d*(d*x+c)^(3/2)/b-(b^(1/2)*c-a^(1/2)*d)^(5/2)*a 
rctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(1/2)/b^(7/4)+ 
(b^(1/2)*c+a^(1/2)*d)^(5/2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/ 
2)*d)^(1/2))/a^(1/2)/b^(7/4)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=-\frac {2 b d \sqrt {c+d x} (7 c+d x)+\frac {3 \left (\sqrt {b} c+\sqrt {a} d\right )^2 \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a}}+\frac {3 \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{3 b^2} \] Input:

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

Output:

-1/3*(2*b*d*Sqrt[c + d*x]*(7*c + d*x) + (3*(Sqrt[b]*c + Sqrt[a]*d)^2*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)])/Sqrt[a] + (3*Sqrt[b]*(Sqrt[b]*c - Sqrt 
[a]*d)^3*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]* 
c - Sqrt[a]*d)])/(Sqrt[a]*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/b^2
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {481, 25, 653, 25, 27, 654, 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.

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

\(\Big \downarrow \) 481

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 653

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 d \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} d}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} b^{3/4} d}\right )-4 c d \sqrt {c+d x}}{b}-\frac {2 d (c+d x)^{3/2}}{3 b}\)

Input:

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

Output:

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

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 481

Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c 
 + d*x)^(n - 1)/(b*(n - 1))), x] + Simp[1/b   Int[(c + d*x)^(n - 2)*(Simp[b 
*c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] 
 && GtQ[n, 1]
 

rule 653

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), 
 x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c   Int[(d + e*x)^(m 
- 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr 
eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

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]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(-\frac {d \left (\frac {2 \sqrt {d x +c}\, \left (d x +7 c \right )}{3}-\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b}\)	\(216\)
risch	\(-\frac {2 \left (d x +7 c \right ) d \sqrt {d x +c}}{3 b}-2 d \left (\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(225\)
derivativedivides	\(-2 d \left (\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {d x +c}}{b}+\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(228\)
default	\(2 d \left (-\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {d x +c}}{b}+\frac {\left (3 a b c \,d^{2}+c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-3 a b c \,d^{2}-c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )\)	\(231\)

Input:

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

Output:

-d/b*(2/3*(d*x+c)^(1/2)*(d*x+7*c)-(3*a*b*c*d^2+c^3*b^2+(a*b*d^2)^(1/2)*a*d 
^2+3*(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))+(-3*a*b*c*d^2-c 
^3*b^2+(a*b*d^2)^(1/2)*a*d^2+3*(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. 1617 vs. \(2 (121) = 242\).

Time = 0.19 (sec) , antiderivative size = 1617, normalized size of antiderivative = 9.68 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(3*b*sqrt((b^2*c^5 + 10*a*b*c^3*d^2 + 5*a^2*c*d^4 + a*b^3*sqrt((25*b^ 
4*c^8*d^2 + 100*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 20*a^3*b*c^2*d^8 + a 
^4*d^10)/(a*b^7)))/(a*b^3))*log((5*b^4*c^8*d - 14*a^2*b^2*c^4*d^5 + 8*a^3* 
b*c^2*d^7 + a^4*d^9)*sqrt(d*x + c) + (10*a*b^4*c^5*d^2 + 20*a^2*b^3*c^3*d^ 
4 + 2*a^3*b^2*c*d^6 - (a*b^6*c^2 + a^2*b^5*d^2)*sqrt((25*b^4*c^8*d^2 + 100 
*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 20*a^3*b*c^2*d^8 + a^4*d^10)/(a*b^7 
)))*sqrt((b^2*c^5 + 10*a*b*c^3*d^2 + 5*a^2*c*d^4 + a*b^3*sqrt((25*b^4*c^8* 
d^2 + 100*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 20*a^3*b*c^2*d^8 + a^4*d^1 
0)/(a*b^7)))/(a*b^3))) - 3*b*sqrt((b^2*c^5 + 10*a*b*c^3*d^2 + 5*a^2*c*d^4 
+ a*b^3*sqrt((25*b^4*c^8*d^2 + 100*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 2 
0*a^3*b*c^2*d^8 + a^4*d^10)/(a*b^7)))/(a*b^3))*log((5*b^4*c^8*d - 14*a^2*b 
^2*c^4*d^5 + 8*a^3*b*c^2*d^7 + a^4*d^9)*sqrt(d*x + c) - (10*a*b^4*c^5*d^2 
+ 20*a^2*b^3*c^3*d^4 + 2*a^3*b^2*c*d^6 - (a*b^6*c^2 + a^2*b^5*d^2)*sqrt((2 
5*b^4*c^8*d^2 + 100*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 20*a^3*b*c^2*d^8 
 + a^4*d^10)/(a*b^7)))*sqrt((b^2*c^5 + 10*a*b*c^3*d^2 + 5*a^2*c*d^4 + a*b^ 
3*sqrt((25*b^4*c^8*d^2 + 100*a*b^3*c^6*d^4 + 110*a^2*b^2*c^4*d^6 + 20*a^3* 
b*c^2*d^8 + a^4*d^10)/(a*b^7)))/(a*b^3))) + 3*b*sqrt((b^2*c^5 + 10*a*b*c^3 
*d^2 + 5*a^2*c*d^4 - a*b^3*sqrt((25*b^4*c^8*d^2 + 100*a*b^3*c^6*d^4 + 110* 
a^2*b^2*c^4*d^6 + 20*a^3*b*c^2*d^8 + a^4*d^10)/(a*b^7)))/(a*b^3))*log((5*b 
^4*c^8*d - 14*a^2*b^2*c^4*d^5 + 8*a^3*b*c^2*d^7 + a^4*d^9)*sqrt(d*x + c...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (121) = 242\).

Time = 0.18 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.49 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=-\frac {{\left (\sqrt {a b} b^{4} c^{4} d + 3 \, \sqrt {a b} a b^{3} c^{2} d^{3} - {\left (3 \, \sqrt {a b} a b c^{2} d + \sqrt {a b} a^{2} d^{3}\right )} b^{2} d^{2} + 2 \, {\left (a b^{3} c^{3} d - a^{2} b^{2} c d^{3}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{4} c + \sqrt {b^{8} c^{2} - {\left (b^{4} c^{2} - a b^{3} d^{2}\right )} b^{4}}}{b^{4}}}}\right )}{{\left (a b^{4} c - \sqrt {a b} a b^{3} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} + \frac {{\left (\sqrt {a b} b^{4} c^{4} d + 3 \, \sqrt {a b} a b^{3} c^{2} d^{3} - {\left (3 \, \sqrt {a b} a b c^{2} d + \sqrt {a b} a^{2} d^{3}\right )} b^{2} d^{2} - 2 \, {\left (a b^{3} c^{3} d - a^{2} b^{2} c d^{3}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{4} c - \sqrt {b^{8} c^{2} - {\left (b^{4} c^{2} - a b^{3} d^{2}\right )} b^{4}}}{b^{4}}}}\right )}{{\left (a b^{4} c + \sqrt {a b} a b^{3} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} - \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{2} d + 6 \, \sqrt {d x + c} b^{2} c d\right )}}{3 \, b^{3}} \] Input:

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

Output:

-(sqrt(a*b)*b^4*c^4*d + 3*sqrt(a*b)*a*b^3*c^2*d^3 - (3*sqrt(a*b)*a*b*c^2*d 
 + sqrt(a*b)*a^2*d^3)*b^2*d^2 + 2*(a*b^3*c^3*d - a^2*b^2*c*d^3)*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))/((a*b^4*c - sqrt(a*b)*a*b^3*d)*sqrt(-b^2*c - sqrt(a*b)*b*d 
)*abs(d)) + (sqrt(a*b)*b^4*c^4*d + 3*sqrt(a*b)*a*b^3*c^2*d^3 - (3*sqrt(a*b 
)*a*b*c^2*d + sqrt(a*b)*a^2*d^3)*b^2*d^2 - 2*(a*b^3*c^3*d - a^2*b^2*c*d^3) 
*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))/((a*b^4*c + sqrt(a*b)*a*b^3*d)*sqrt(-b^2*c + sq 
rt(a*b)*b*d)*abs(d)) - 2/3*((d*x + c)^(3/2)*b^2*d + 6*sqrt(d*x + c)*b^2*c* 
d)/b^3
 

Mupad [B] (verification not implemented)

Time = 8.78 (sec) , antiderivative size = 3385, normalized size of antiderivative = 20.27 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((a^3*d^8*(c + d*x)^(1/2)*((d^5*(a^3*b^7)^(1/2))/(4*b^7) + c^5/(4*a* 
b) + (5*c^3*d^2)/(2*b^2) + (5*a*c*d^4)/(4*b^3) + (5*c^4*d*(a^3*b^7)^(1/2)) 
/(4*a^2*b^5) + (5*c^2*d^3*(a^3*b^7)^(1/2))/(2*a*b^6))^(1/2)*32i)/((16*a^4* 
d^11)/b^2 + 64*a^2*c^4*d^7 - 80*b^2*c^8*d^3 + (160*a^3*c^2*d^9)/b - 160*a* 
b*c^6*d^5 - (160*c^5*d^6*(a^3*b^7)^(1/2))/b^3 + (288*a*c^3*d^8*(a^3*b^7)^( 
1/2))/b^4 + (32*a^2*c*d^10*(a^3*b^7)^(1/2))/b^5 - (160*c^7*d^4*(a^3*b^7)^( 
1/2))/(a*b^2)) - (c^3*d^5*(a^3*b^7)^(1/2)*(c + d*x)^(1/2)*((d^5*(a^3*b^7)^ 
(1/2))/(4*b^7) + c^5/(4*a*b) + (5*c^3*d^2)/(2*b^2) + (5*a*c*d^4)/(4*b^3) + 
 (5*c^4*d*(a^3*b^7)^(1/2))/(4*a^2*b^5) + (5*c^2*d^3*(a^3*b^7)^(1/2))/(2*a* 
b^6))^(1/2)*320i)/(16*a^4*d^11 - 80*b^4*c^8*d^3 - 160*a*b^3*c^6*d^5 + 160* 
a^3*b*c^2*d^9 + 64*a^2*b^2*c^4*d^7 - (160*c^7*d^4*(a^3*b^7)^(1/2))/a - (16 
0*c^5*d^6*(a^3*b^7)^(1/2))/b + (288*a*c^3*d^8*(a^3*b^7)^(1/2))/b^2 + (32*a 
^2*c*d^10*(a^3*b^7)^(1/2))/b^3) - (c^5*d^3*(a^3*b^7)^(1/2)*(c + d*x)^(1/2) 
*((d^5*(a^3*b^7)^(1/2))/(4*b^7) + c^5/(4*a*b) + (5*c^3*d^2)/(2*b^2) + (5*a 
*c*d^4)/(4*b^3) + (5*c^4*d*(a^3*b^7)^(1/2))/(4*a^2*b^5) + (5*c^2*d^3*(a^3* 
b^7)^(1/2))/(2*a*b^6))^(1/2)*160i)/((16*a^5*d^11)/b + 160*a^4*c^2*d^9 - 80 
*a*b^3*c^8*d^3 + 64*a^3*b*c^4*d^7 - 160*a^2*b^2*c^6*d^5 - (160*c^7*d^4*(a^ 
3*b^7)^(1/2))/b - (160*a*c^5*d^6*(a^3*b^7)^(1/2))/b^2 + (32*a^3*c*d^10*(a^ 
3*b^7)^(1/2))/b^4 + (288*a^2*c^3*d^8*(a^3*b^7)^(1/2))/b^3) + (a*b^2*c^4*d^ 
4*(c + d*x)^(1/2)*((d^5*(a^3*b^7)^(1/2))/(4*b^7) + c^5/(4*a*b) + (5*c^3...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.54 \[ \int \frac {(c+d x)^{5/2}}{a-b x^2} \, dx=\frac {-6 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a \,d^{2}-6 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b \,c^{2}+12 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a c d -3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{2}-3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{2}+3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{2}+3 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b \,c^{2}-6 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c d +6 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c d -28 \sqrt {d x +c}\, a b c d -4 \sqrt {d x +c}\, a b \,d^{2} x}{6 a \,b^{2}} \] Input:

int((d*x+c)^(5/2)/(-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)))*a*d**2 - 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)))*b 
*c**2 + 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(s 
qrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*c*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)) 
*a*d**2 - 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))*b*c**2 + 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))*a* 
d**2 + 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))*b*c**2 - 6*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))*a*c*d 
 + 6*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))*a*c*d - 28*sqrt(c + d*x)*a*b*c*d - 4*sqrt(c + 
d*x)*a*b*d**2*x)/(6*a*b**2)