\(\int \frac {(c+d x)^{5/2}}{(a-b x^2)^{3/2}} \, dx\) [1548]

Optimal result

Integrand size = 22, antiderivative size = 355 \[ \int \frac {(c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {(a d+b c x) (c+d x)^{3/2}}{a b \sqrt {a-b x^2}}+\frac {c d \sqrt {c+d x} \sqrt {a-b x^2}}{a b}+\frac {\left (b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(b*c*x+a*d)*(d*x+c)^(3/2)/a/b/(-b*x^2+a)^(1/2)+c*d*(d*x+c)^(1/2)*(-b*x^2+a 
)^(1/2)/a/b+(3*a*d^2+b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2* 
(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)* 
d))^(1/2))/a^(1/2)/b^(3/2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/( 
-b*x^2+a)^(1/2)-c*(-a*d^2+b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^( 
1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2 
^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/b^(3/2)/(d*x+c)^(1 
/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {b (c+d x) \left (b c^2 x+a d (2 c+d x)\right )}{-a+b x^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2+3 a d^2\right ) \left (a-b x^2\right )+i \sqrt {b} \left (b^{3/2} c^3-\sqrt {a} b c^2 d+3 a \sqrt {b} c d^2-3 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (b c^2-4 \sqrt {a} \sqrt {b} c d+3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{a b^2 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(-((b*(c + d*x)*(b*c^2*x + a*d*(2*c + d*x)))/(-a + b*x^2) 
) - (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 + 3*a*d^2)*(a - b*x^2) + I* 
Sqrt[b]*(b^(3/2)*c^3 - Sqrt[a]*b*c^2*d + 3*a*Sqrt[b]*c*d^2 - 3*a^(3/2)*d^3 
)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - 
d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] 
 + I*Sqrt[a]*Sqrt[b]*d*(b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*Sqrt[(d*( 
Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d 
*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/S 
qrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(a*b^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {495, 27, 687, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 495

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

\(\displaystyle \frac {(c+d x)^{3/2} (a d+b c x)}{a b \sqrt {a-b x^2}}-\frac {3 d \left (\frac {1}{3} \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{2 a b}\)

Input:

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

Output:

((a*d + b*c*x)*(c + d*x)^(3/2))/(a*b*Sqrt[a - b*x^2]) - (3*d*((-2*c*Sqrt[c 
 + d*x]*Sqrt[a - b*x^2])/3 + ((-2*Sqrt[a]*(b*c^2 + 3*a*d^2)*Sqrt[c + d*x]* 
Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2] 
], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/( 
Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*Sq 
rt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellipt 
icF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt 
[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/3))/(2*a*b)
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 495

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(a*d - b*c*x)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - 
 Simp[1/(2*a*b*(p + 1))   Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* 
d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ 
{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, 
 n, p, x]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(677\) vs. \(2(293)=586\).

Time = 2.65 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.91

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {\left (a \,d^{2}+b \,c^{2}\right ) x}{2 a \,b^{2}}+\frac {c d}{b^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (-\frac {4 c \,d^{2}}{b}+\frac {c \left (3 a \,d^{2}+b \,c^{2}\right )}{a b}-\frac {c \left (a \,d^{2}+b \,c^{2}\right )}{b a}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {d^{3}}{b}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) d}{2 a b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(678\)
default	\(\text {Expression too large to display}\)	\(1093\)

Input:

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

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) 
*(1/2*(a*d^2+b*c^2)/a/b^2*x+c*d/b^2)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-4* 
c*d^2/b+c*(3*a*d^2+b*c^2)/a/b-1/b*c*(a*d^2+b*c^2)/a)*(c/d-1/b*(a*b)^(1/2)) 
*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b 
)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x 
^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2 
),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(-d^3/b-1/2*(a* 
d^2+b*c^2)*d/a/b)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1 
/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2) 
)/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d- 
1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/ 
b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF((( 
x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b 
)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (a b c^{3} - 9 \, a^{2} c d^{2} - {\left (b^{2} c^{3} - 9 \, a b c d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (a b c^{2} d + 3 \, a^{2} d^{3} - {\left (b^{2} c^{2} d + 3 \, a b d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (2 \, a b c d^{2} + {\left (b^{2} c^{2} d + a b d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a b^{3} d x^{2} - a^{2} b^{2} d\right )}} \] Input:

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

Output:

1/3*((a*b*c^3 - 9*a^2*c*d^2 - (b^2*c^3 - 9*a*b*c*d^2)*x^2)*sqrt(-b*d)*weie 
rstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/( 
b*d^3), 1/3*(3*d*x + c)/d) + 3*(a*b*c^2*d + 3*a^2*d^3 - (b^2*c^2*d + 3*a*b 
*d^3)*x^2)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27 
*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b 
*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) - 3*(2*a*b*c 
*d^2 + (b^2*c^2*d + a*b*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^3*d*x 
^2 - a^2*b^2*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a 
 - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b 
*d**2*x**4),x)*a**3*c*d**4 + 2*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a 
- b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2* 
x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b*c**3*d**2 + sqrt(a - b*x**2 
)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x** 
2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a*b**2 
*c**5 - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c 
 + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)* 
a*b**2*c*d**3 + sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3) 
/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x 
**5),x)*b**3*c**3*d - 3*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x** 
2)*x**2)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + 
b**2*d*x**5),x)*a**2*b*d**4 + 3*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a 
 - b*x**2)*x**2)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c 
*x**4 + b**2*d*x**5),x)*a*b**2*c**2*d**2 + 2*sqrt(a - b*x**2)*int((sqrt(c 
+ d*x)*sqrt(a - b*x**2))/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 
+ b**2*c*x**4 + b**2*d*x**5),x)*a**2*b*c**2*d**2 - 2*sqrt(a - b*x**2)*int( 
(sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b 
*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**2*c**4 + 3*sqrt(a - b*x**2...