3.31.36 \(\int \frac {(d+c x^2) (a x+\sqrt {-b+a^2 x^2})^{3/4}}{(-b+a^2 x^2)^{5/2}} \, dx\) [3036]

3.31.36.1 Optimal result

Integrand size = 46, antiderivative size = 448 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx=\frac {\left (-53 b^2 c+43 a^2 b d+41 a^2 b c x^2-55 a^4 d x^2\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^3 b^{13/8} \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {\sqrt {-b+a^2 x^2} \left (9 b^2 c x-87 a^2 b d x-41 a^2 b c x^3+55 a^4 d x^3\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^2 b^{13/8} \left (-\sqrt {b}+a x\right )^2 \left (\sqrt {b}+a x\right )^2}+\frac {\left (41 b c-55 a^2 d\right ) \arctan \left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}+\frac {\left (-41 b c+55 a^2 d\right ) \text {arctanh}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {\sqrt [4]{-1} \left (-41 b c+55 a^2 d\right ) \text {arctanh}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {(-1)^{3/4} \left (-41 b c+55 a^2 d\right ) \text {arctanh}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}} \]

output

1/96*(-55*a^4*d*x^2+41*a^2*b*c*x^2+43*a^2*b*d-53*b^2*c)*((a*x+(a^2*x^2-b)^ 
(1/2))/b^(1/2))^(3/4)/a^3/b^(13/8)/(-b^(1/2)+a*x)/(b^(1/2)+a*x)+1/96*(a^2* 
x^2-b)^(1/2)*(55*a^4*d*x^3-41*a^2*b*c*x^3-87*a^2*b*d*x+9*b^2*c*x)*((a*x+(a 
^2*x^2-b)^(1/2))/b^(1/2))^(3/4)/a^2/b^(13/8)/(-b^(1/2)+a*x)^2/(b^(1/2)+a*x 
)^2+1/64*(-55*a^2*d+41*b*c)*arctan(((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4) 
)/a^3/b^(13/8)+1/64*(55*a^2*d-41*b*c)*arctanh(((a*x+(a^2*x^2-b)^(1/2))/b^( 
1/2))^(1/4))/a^3/b^(13/8)-1/64*(-1)^(1/4)*(55*a^2*d-41*b*c)*arctanh((-1)^( 
1/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^3/b^(13/8)-1/64*(-1)^(3/4) 
*(55*a^2*d-41*b*c)*arctanh((-1)^(3/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1 
/4))/a^3/b^(13/8)
 

3.31.36.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.97 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.46 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx=\frac {16 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (273 c \left (\frac {1}{b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2}-\frac {\operatorname {Hypergeometric2F1}\left (\frac {3}{8},2,\frac {11}{8},\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^2}{b}\right )}{b}\right )-20 \left (b c+a^2 d\right ) \left (\frac {16 b-21 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}{4 \left (b-a x \left (a x+\sqrt {-b+a^2 x^2}\right )\right )^3}-\frac {11 \operatorname {Hypergeometric2F1}\left (\frac {3}{8},4,\frac {11}{8},\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^2}{b}\right )}{b^2}\right )\right )}{1365 a^3} \]

input

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

output

(16*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(273*c*((b - (a*x + Sqrt[-b + a^2*x^2 
])^2)^(-1) - Hypergeometric2F1[3/8, 2, 11/8, (a*x + Sqrt[-b + a^2*x^2])^2/ 
b]/b) - 20*(b*c + a^2*d)*((16*b - 21*a*x*(a*x + Sqrt[-b + a^2*x^2]))/(4*(b 
 - a*x*(a*x + Sqrt[-b + a^2*x^2]))^3) - (11*Hypergeometric2F1[3/8, 4, 11/8 
, (a*x + Sqrt[-b + a^2*x^2])^2/b])/b^2)))/(1365*a^3)
 

3.31.36.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1077\) vs. \(2(448)=896\).

Time = 2.15 (sec) , antiderivative size = 1077, normalized size of antiderivative = 2.40, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {11 c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {8 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {41 c \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {41 c \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}+\frac {41 c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^3 b^{5/8}}-\frac {55 d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a^3 b^{5/8}}+\frac {55 d \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a b^{13/8}}-\frac {41 c \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 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 )}{128 \sqrt {2} a^3 b^{5/8}}+\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}+\frac {41 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 )}{128 \sqrt {2} a^3 b^{5/8}}-\frac {55 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 )}{128 \sqrt {2} a b^{13/8}}\)

input

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

output

(8*b*c*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(3*a^3*(b - (a*x + Sqrt[-b + a^2 
*x^2])^2)^3) + (8*d*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(3*a*(b - (a*x + Sq 
rt[-b + a^2*x^2])^2)^3) - (11*d*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(6*a*(b 
- (a*x + Sqrt[-b + a^2*x^2])^2)^2) - (11*c*(a*x + Sqrt[-b + a^2*x^2])^(11/ 
4))/(6*a^3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^2) + (41*c*(a*x + Sqrt[-b + 
a^2*x^2])^(3/4))/(48*a^3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + (11*d*(a*x 
+ Sqrt[-b + a^2*x^2])^(3/4))/(16*a*b*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + 
 (41*c*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a^3*b^(5/8)) 
- (55*d*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a*b^(13/8)) 
+ (41*c*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(6 
4*Sqrt[2]*a^3*b^(5/8)) - (55*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^ 
2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a*b^(13/8)) - (41*c*ArcTan[1 + (Sqrt[2]*( 
a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a^3*b^(5/8)) + (55* 
d*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt 
[2]*a*b^(13/8)) - (41*c*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]) 
/(64*a^3*b^(5/8)) + (55*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8) 
])/(64*a*b^(13/8)) - (41*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]]])/(128*Sqrt[2]*a^3*b^(5/8 
)) + (55*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]]])/(128*Sqrt[2]*a*b^(13/8)) + (41*c*Log...
 

3.31.36.3.1 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] 
]
 

3.31.36.4 Maple [F]

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

input

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

output

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

3.31.36.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 2886, normalized size of antiderivative = 6.44 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

input

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

output

-1/768*(3*sqrt(2)*(-(I - 1)*a^7*b^2*x^4 + (2*I - 2)*a^5*b^3*x^2 - (I - 1)* 
a^3*b^4)*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 130287 
2392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 18100235 
47543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212 
700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8 
)/(a^24*b^13))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^9*b^5*((83733937890625*a^ 
16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 
- 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1 
079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 856918 
80507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(3/8) + (1663 
75*a^6*d^3 - 372075*a^4*b*c*d^2 + 277365*a^2*b^2*c^2*d - 68921*b^3*c^3)*(a 
*x + sqrt(a^2*x^2 - b))^(1/4)) + 3*sqrt(2)*((I + 1)*a^7*b^2*x^4 - (2*I + 2 
)*a^5*b^3*x^2 + (I + 1)*a^3*b^4)*((83733937890625*a^16*d^8 - 4993587568750 
00*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^1 
0*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^ 
5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d 
 + 7984925229121*b^8*c^8)/(a^24*b^13))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^ 
9*b^5*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 130287239 
2937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 18100235475 
43750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212...
 

3.31.36.6 Sympy [F]

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

input

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

output

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

3.31.36.7 Maxima [F]

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

input

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

output

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

3.31.36.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.31.36.9 Mupad [F(-1)]

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

input

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

output

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