\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2)}{x^4 \sqrt {c+d x}} \, dx\) [182]

Optimal result

Integrand size = 35, antiderivative size = 693 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4 \sqrt {c+d x}} \, dx=-\frac {2 b C \sqrt {c+d x} \sqrt {a-b x^2}}{3 d}-\frac {a A \sqrt {c+d x} \sqrt {a-b x^2}}{3 c x^3}-\frac {a (6 B c-5 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{12 c^2 x^2}-\frac {\left (6 a c (4 c C-3 B d)-A \left (32 b c^2-15 a d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{24 c^3 x}+\frac {\sqrt {a} \sqrt {b} \left (16 b c^2 \left (2 c^2 C-3 B c d-2 A d^2\right )+3 a d^2 \left (8 c^2 C-6 B c d+5 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{24 c^3 d^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \sqrt {b} \left (a d^2 \left (56 c^2 C+6 B c d-5 A d^2\right )-16 b c^2 \left (2 c^2 C-3 B c d+A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{24 c^2 d^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {a \left (12 b c^2 (2 B c-A d)+a d \left (8 c^2 C-6 B c d+5 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{8 c^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/3*b*C*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d-1/3*a*A*(d*x+c)^(1/2)*(-b*x^2+a) 
^(1/2)/c/x^3-1/12*a*(-5*A*d+6*B*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c^2/x^2- 
1/24*(6*a*c*(-3*B*d+4*C*c)-A*(-15*a*d^2+32*b*c^2))*(d*x+c)^(1/2)*(-b*x^2+a 
)^(1/2)/c^3/x+1/24*a^(1/2)*b^(1/2)*(16*b*c^2*(-2*A*d^2-3*B*c*d+2*C*c^2)+3* 
a*d^2*(5*A*d^2-6*B*c*d+8*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellipt 
icE(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))/c^3/d^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a 
)^(1/2)+1/24*a^(1/2)*b^(1/2)*(a*d^2*(-5*A*d^2+6*B*c*d+56*C*c^2)-16*b*c^2*( 
A*d^2-3*B*c*d+2*C*c^2))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/ 
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))/c^2/d^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+1 
/8*a*(12*b*c^2*(-A*d+2*B*c)+a*d*(5*A*d^2-6*B*c*d+8*C*c^2))*((d*x+c)/(c+a^( 
1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^ 
(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/c^ 
3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.90 (sec) , antiderivative size = 2192, normalized size of antiderivative = 3.16 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4 \sqrt {c+d x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/24*(Sqrt[c + d*x]*Sqrt[a - b*x^2]*(16*b*c^2*x^2*(-2*A*d + c*C*x) + a*d* 
(6*c*x*(2*B*c + 4*c*C*x - 3*B*d*x) + A*(8*c^2 - 10*c*d*x + 15*d^2*x^2))))/ 
(c^3*d*x^3) - (32*b^2*c^7*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 48*b^2*B*c^6* 
d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 32*A*b^2*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/ 
Sqrt[b]] - 8*a*b*c^5*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 30*a*b*B*c^4*d 
^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 47*a*A*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]] - 24*a^2*c^3*C*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 18*a^2*B*c^2 
*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 15*a^2*A*c*d^6*Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]] - 64*b^2*c^6*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 96*b^2 
*B*c^5*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 64*A*b^2*c^4*d^2*Sqrt[ 
-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 48*a*b*c^4*C*d^2*Sqrt[-c + (Sqrt[a]* 
d)/Sqrt[b]]*(c + d*x) + 36*a*b*B*c^3*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c 
 + d*x) - 30*a*A*b*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 32*b 
^2*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 48*b^2*B*c^4*d*Sqrt[ 
-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 32*A*b^2*c^3*d^2*Sqrt[-c + (Sqrt[a 
]*d)/Sqrt[b]]*(c + d*x)^2 + 24*a*b*c^3*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] 
]*(c + d*x)^2 - 18*a*b*B*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^ 
2 + 15*a*A*b*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + I*Sqrt[b]* 
c*(Sqrt[b]*c - Sqrt[a]*d)*(16*b*c^2*(-2*c^2*C + 3*B*c*d + 2*A*d^2) - 3*a*d 
^2*(8*c^2*C - 6*B*c*d + 5*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2355

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

\(\Big \downarrow \) 638

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7296

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2011

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

\(\Big \downarrow \) 2091

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

Input:

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

Output:

$Aborted
 

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 638

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ 
[{a, b, c, d, e, m, n, p}, x]
 

rule 2011

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x 
, a + b*x])
 

rule 2091

Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex 
pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x 
] && TrinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
 

rule 2248

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 
+ c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; 
 FreeQ[{a, b, c, d, e, f, m}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && In 
tegerQ[q]
 

rule 2355

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* 
x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x]   I 
nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} 
, x] && PolynomialQ[Px, x] && LtQ[n, 0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

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

rule 7296

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst]]
 

Maple [A] (verified)

Time = 6.19 (sec) , antiderivative size = 1052, normalized size of antiderivative = 1.52

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1052\)
risch	\(\text {Expression too large to display}\)	\(1717\)
default	\(\text {Expression too large to display}\)	\(5353\)

Input:

int((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^(1/2),x,method=_RETURNVERBO 
SE)
                                                                                    
                                                                                    
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-1/3*A*a/x^3/c* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/12*a*(5*A*d-6*B*c)/c^2*(-b*d*x^3-b*c* 
x^2+a*d*x+a*c)^(1/2)/x^2-1/24/c^3*(15*A*a*d^2-32*A*b*c^2-18*B*a*c*d+24*C*a 
*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x-2/3*C*b/d*(-b*d*x^3-b*c*x^2+a*d 
*x+a*c)^(1/2)+2*(A*b^2-5/3*C*b*a-1/24*b*d*a*(5*A*d-6*B*c)/c^2)*(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*(B*b^ 
2-1/48*b*d*(15*A*a*d^2-32*A*b*c^2-18*B*a*c*d+24*C*a*c^2)/c^3-2/3*C*b^2/d*c 
)*(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)) 
)+1/8*a*(5*A*a*d^3-12*A*b*c^2*d-6*B*a*c*d^2+24*B*b*c^3+8*C*a*c^2*d)/c^4*(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)*d*EllipticPi(((x+c/d)/(c...
 

Fricas [F]

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

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral(-(C*b*x^4 + B*b*x^3 - B*a*x - (C*a - A*b)*x^2 - A*a)*sqrt(-b*x^2 
+ a)*sqrt(d*x + c)/(d*x^5 + c*x^4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^(1/2),x, algorithm="m 
axima")
 

Output:

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

Giac [F]

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

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4/(d*x+c)^(1/2),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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