\(\int \frac {(d+c x^2) (a x+\sqrt {-b+a^2 x^2})^{5/4}}{(-b+a^2 x^2)^{3/2}} \, dx\) [2982]

Optimal result

Integrand size = 46, antiderivative size = 384 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-b+a^2 x^2} \left (-9 b^{9/8} c-5 a^2 \sqrt [8]{b} d+4 a^2 \sqrt [8]{b} c x^2\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^3 \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {4 \sqrt [8]{b} c x \sqrt [4]{\frac {a x}{\sqrt {b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{5 a^2}-\frac {5 \left (b c+a^2 d\right ) \arctan \left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \left (b c+a^2 d\right ) \text {arctanh}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 (-1)^{3/4} \left (b c+a^2 d\right ) \text {arctanh}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}}-\frac {5 \sqrt [4]{-1} \left (b c+a^2 d\right ) \text {arctanh}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{2 a^3 b^{3/8}} \] Output:

1/5*(a^2*x^2-b)^(1/2)*(-9*b^(9/8)*c-5*a^2*b^(1/8)*d+4*a^2*b^(1/8)*c*x^2)*( 
(a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4)/a^3/(-b^(1/2)+a*x)/(b^(1/2)+a*x)+4/ 
5*b^(1/8)*c*x*(a*x/b^(1/2)+(a^2*x^2-b)^(1/2)/b^(1/2))^(1/4)/a^2-5/2*(a^2*d 
+b*c)*arctan(((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^3/b^(3/8)-5/2*(a^2 
*d+b*c)*arctanh(((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^3/b^(3/8)-5/2*( 
-1)^(3/4)*(a^2*d+b*c)*arctanh((-1)^(1/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2)) 
^(1/4))/a^3/b^(3/8)-5/2*(-1)^(1/4)*(a^2*d+b*c)*arctanh((-1)^(3/4)*((a*x+(a 
^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^3/b^(3/8)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(13171\) vs. \(2(384)=768\).

Time = 40.23 (sec) , antiderivative size = 13171, normalized size of antiderivative = 34.30 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3 
/2),x]
 

Output:

Result too large to show
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(869\) vs. \(2(384)=768\).

Time = 2.12 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{5 a^3}+\frac {2 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {2 d \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}}{a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a^3}-\frac {5 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a^3}+\frac {5 d \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} a b^{3/8}}-\frac {5 b^{5/8} c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a^3}-\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 a b^{3/8}}-\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}-\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}}+\frac {5 b^{5/8} c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a^3}+\frac {5 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{4 \sqrt {2} a b^{3/8}}\)

Input:

Int[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2),x]
 

Output:

(4*c*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(5*a^3) + (2*b*c*(a*x + Sqrt[-b + a 
^2*x^2])^(5/4))/(a^3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + (2*d*(a*x + Sqr 
t[-b + a^2*x^2])^(5/4))/(a*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) - (5*b^(5/8 
)*c*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a^3) - (5*d*ArcTa 
n[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a*b^(3/8)) - (5*b^(5/8)*c* 
ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2] 
*a^3) - (5*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8) 
])/(2*Sqrt[2]*a*b^(3/8)) + (5*b^(5/8)*c*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b 
 + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2]*a^3) + (5*d*ArcTan[1 + (Sqrt[2]*( 
a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(2*Sqrt[2]*a*b^(3/8)) - (5*b^(5 
/8)*c*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a^3) - (5*d*Ar 
cTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*a*b^(3/8)) - (5*b^(5/8 
)*c*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[ 
a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]*a^3) - (5*d*Log[b^(1/4) - Sqrt[2]*b 
^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]) 
/(4*Sqrt[2]*a*b^(3/8)) + (5*b^(5/8)*c*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + 
 Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]*a 
^3) + (5*d*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) 
+ Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]*a*b^(3/8))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x)
 

Output:

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 2872, normalized size of antiderivative = 7.48 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, alg 
orithm="fricas")
 

Output:

-1/40*(25*sqrt(2)*(-(I + 1)*a^5*x^2 + (I + 1)*a^3*b)*((a^16*d^8 + 8*a^14*b 
*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 
56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24 
*b^3))^(1/8)*log((3125/2*I + 3125/2)*sqrt(2)*a^15*b^2*((a^16*d^8 + 8*a^14* 
b*c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 
 56*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^2 
4*b^3))^(5/8) + 3125*(a^10*d^5 + 5*a^8*b*c*d^4 + 10*a^6*b^2*c^2*d^3 + 10*a 
^4*b^3*c^3*d^2 + 5*a^2*b^4*c^4*d + b^5*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4 
)) + 25*sqrt(2)*((I - 1)*a^5*x^2 - (I - 1)*a^3*b)*((a^16*d^8 + 8*a^14*b*c* 
d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56* 
a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^ 
3))^(1/8)*log(-(3125/2*I - 3125/2)*sqrt(2)*a^15*b^2*((a^16*d^8 + 8*a^14*b* 
c*d^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 5 
6*a^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24* 
b^3))^(5/8) + 3125*(a^10*d^5 + 5*a^8*b*c*d^4 + 10*a^6*b^2*c^2*d^3 + 10*a^4 
*b^3*c^3*d^2 + 5*a^2*b^4*c^4*d + b^5*c^5)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) 
 + 25*sqrt(2)*(-(I - 1)*a^5*x^2 + (I - 1)*a^3*b)*((a^16*d^8 + 8*a^14*b*c*d 
^7 + 28*a^12*b^2*c^2*d^6 + 56*a^10*b^3*c^3*d^5 + 70*a^8*b^4*c^4*d^4 + 56*a 
^6*b^5*c^5*d^3 + 28*a^4*b^6*c^6*d^2 + 8*a^2*b^7*c^7*d + b^8*c^8)/(a^24*b^3 
))^(1/8)*log((3125/2*I - 3125/2)*sqrt(2)*a^15*b^2*((a^16*d^8 + 8*a^14*b...
 

Sympy [F]

\[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {5}{4}} \left (c x^{2} + d\right )}{\left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((c*x**2+d)*(a*x+(a**2*x**2-b)**(1/2))**(5/4)/(a**2*x**2-b)**(3/2 
),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + d\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {5}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{5/4}\,\left (c\,x^2+d\right )}{{\left (a^2\,x^2-b\right )}^{3/2}} \,d x \] Input:

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(a^2*x^2 - b)^(3/2),x)
 

Output:

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(a^2*x^2 - b)^(3/2), x 
)
 

Reduce [F]

\[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/(a^2*x^2-b)^(3/2),x)
 

Output:

(56*sqrt(a**2*x**2 - b)*(sqrt(a**2*x**2 - b) + a*x)**(1/4)*a**2*b*c*x**2 - 
 160*sqrt(a**2*x**2 - b)*(sqrt(a**2*x**2 - b) + a*x)**(1/4)*a**2*b*d - 180 
*sqrt(a**2*x**2 - b)*(sqrt(a**2*x**2 - b) + a*x)**(1/4)*b**2*c - 75*int((s 
qrt(a**2*x**2 - b) + a*x)**(1/4)/(a**4*x**4 - 2*a**2*b*x**2 + b**2),x)*a** 
5*b**2*d*x**2 - 45*int((sqrt(a**2*x**2 - b) + a*x)**(1/4)/(a**4*x**4 - 2*a 
**2*b*x**2 + b**2),x)*a**3*b**3*c*x**2 + 75*int((sqrt(a**2*x**2 - b) + a*x 
)**(1/4)/(a**4*x**4 - 2*a**2*b*x**2 + b**2),x)*a**3*b**3*d + 45*int((sqrt( 
a**2*x**2 - b) + a*x)**(1/4)/(a**4*x**4 - 2*a**2*b*x**2 + b**2),x)*a*b**4* 
c - 200*int(((sqrt(a**2*x**2 - b) + a*x)**(1/4)*x**5)/(4*sqrt(a**2*x**2 - 
b)*a**6*x**6 - 9*sqrt(a**2*x**2 - b)*a**4*b*x**4 + 6*sqrt(a**2*x**2 - b)*a 
**2*b**2*x**2 - sqrt(a**2*x**2 - b)*b**3 + 4*a**7*x**7 - 11*a**5*b*x**5 + 
10*a**3*b**2*x**3 - 3*a*b**3*x),x)*a**10*b*d*x**2 - 134*int(((sqrt(a**2*x* 
*2 - b) + a*x)**(1/4)*x**5)/(4*sqrt(a**2*x**2 - b)*a**6*x**6 - 9*sqrt(a**2 
*x**2 - b)*a**4*b*x**4 + 6*sqrt(a**2*x**2 - b)*a**2*b**2*x**2 - sqrt(a**2* 
x**2 - b)*b**3 + 4*a**7*x**7 - 11*a**5*b*x**5 + 10*a**3*b**2*x**3 - 3*a*b* 
*3*x),x)*a**8*b**2*c*x**2 + 200*int(((sqrt(a**2*x**2 - b) + a*x)**(1/4)*x* 
*5)/(4*sqrt(a**2*x**2 - b)*a**6*x**6 - 9*sqrt(a**2*x**2 - b)*a**4*b*x**4 + 
 6*sqrt(a**2*x**2 - b)*a**2*b**2*x**2 - sqrt(a**2*x**2 - b)*b**3 + 4*a**7* 
x**7 - 11*a**5*b*x**5 + 10*a**3*b**2*x**3 - 3*a*b**3*x),x)*a**8*b**2*d + 1 
34*int(((sqrt(a**2*x**2 - b) + a*x)**(1/4)*x**5)/(4*sqrt(a**2*x**2 - b)...