\(\int \frac {x^7}{(c+d x)^2 (a+b x^2)^{5/2}} \, dx\) [1285]

Optimal result

Integrand size = 22, antiderivative size = 295 \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {a^3 \left (b c^2-a d^2-2 b c d x\right )}{3 b^3 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}-\frac {a^2 \left (3 \left (3 b^2 c^4-3 a b c^2 d^2-2 a^2 d^4\right )-4 b c d \left (5 b c^2+2 a d^2\right ) x\right )}{3 b^3 \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^3 d^2}+\frac {c^7 \sqrt {a+b x^2}}{d^2 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {2 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2} d^3}-\frac {c^6 \left (2 b c^2+7 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \left (b c^2+a d^2\right )^{7/2}} \] Output:

1/3*a^3*(-2*b*c*d*x-a*d^2+b*c^2)/b^3/(a*d^2+b*c^2)^2/(b*x^2+a)^(3/2)-1/3*a 
^2*(-6*a^2*d^4-9*a*b*c^2*d^2+9*b^2*c^4-4*b*c*d*(2*a*d^2+5*b*c^2)*x)/b^3/(a 
*d^2+b*c^2)^3/(b*x^2+a)^(1/2)+(b*x^2+a)^(1/2)/b^3/d^2+c^7*(b*x^2+a)^(1/2)/ 
d^2/(a*d^2+b*c^2)^3/(d*x+c)-2*c*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2) 
/d^3-c^6*(7*a*d^2+2*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2 
+a)^(1/2))/d^3/(a*d^2+b*c^2)^(7/2)
 

Mathematica [A] (verified)

Time = 12.35 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.36 \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {d \left (8 a^5 d^6 (c+d x)+3 b^5 c^6 x^4 (2 c+d x)+6 a^4 b d^4 \left (3 c^3+4 c^2 d x+3 c d^2 x^2+2 d^3 x^3\right )+3 a b^4 c^4 x^2 \left (4 c^3+2 c^2 d x+3 c d^2 x^2+3 d^3 x^3\right )+a^3 b^2 d^2 \left (c^5+19 c^4 d x+45 c^3 d^2 x^2+35 c^2 d^3 x^3+11 c d^4 x^4+3 d^5 x^5\right )+a^2 b^3 c^2 \left (6 c^5+3 c^4 d x+9 c^3 d^2 x^2+29 c^2 d^3 x^3+29 c d^4 x^4+9 d^5 x^5\right )\right )}{b^3 \left (b c^2+a d^2\right )^3 (c+d x) \left (a+b x^2\right )^{3/2}}-\frac {6 \sqrt {a} c \sqrt {1+\frac {b x^2}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a+b x^2}}-\frac {3 c^6 \left (2 b c^2+7 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{7/2}}}{3 d^3} \] Input:

Integrate[x^7/((c + d*x)^2*(a + b*x^2)^(5/2)),x]
 

Output:

((d*(8*a^5*d^6*(c + d*x) + 3*b^5*c^6*x^4*(2*c + d*x) + 6*a^4*b*d^4*(3*c^3 
+ 4*c^2*d*x + 3*c*d^2*x^2 + 2*d^3*x^3) + 3*a*b^4*c^4*x^2*(4*c^3 + 2*c^2*d* 
x + 3*c*d^2*x^2 + 3*d^3*x^3) + a^3*b^2*d^2*(c^5 + 19*c^4*d*x + 45*c^3*d^2* 
x^2 + 35*c^2*d^3*x^3 + 11*c*d^4*x^4 + 3*d^5*x^5) + a^2*b^3*c^2*(6*c^5 + 3* 
c^4*d*x + 9*c^3*d^2*x^2 + 29*c^2*d^3*x^3 + 29*c*d^4*x^4 + 9*d^5*x^5)))/(b^ 
3*(b*c^2 + a*d^2)^3*(c + d*x)*(a + b*x^2)^(3/2)) - (6*Sqrt[a]*c*Sqrt[1 + ( 
b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(5/2)*Sqrt[a + b*x^2]) - (3*c^6 
*(2*b*c^2 + 7*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b 
*x^2])])/(b*c^2 + a*d^2)^(7/2))/(3*d^3)
 

Rubi [A] (verified)

Time = 3.25 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {601, 25, 2178, 27, 2182, 2185, 27, 719, 224, 219, 488, 219}

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.

\(\displaystyle \int \frac {x^7}{\left (a+b x^2\right )^{5/2} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2178

\(\displaystyle \frac {-\frac {\int \frac {3 \left (\frac {2 c^3 d \left (3 b c^2+a d^2\right ) a^4}{b \left (b c^2+a d^2\right )^3}+\frac {c^2 \left (2 b^2 c^4+9 a b d^2 c^2+3 a^2 d^4\right ) x a^3}{b \left (b c^2+a d^2\right )^3}-\frac {x^3 a^2}{b}\right )}{(c+d x)^2 \sqrt {b x^2+a}}dx}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {\frac {2 c^3 d \left (3 b c^2+a d^2\right ) a^4}{b \left (b c^2+a d^2\right )^3}+\frac {c^2 \left (2 b^2 c^4+9 a b d^2 c^2+3 a^2 d^4\right ) x a^3}{b \left (b c^2+a d^2\right )^3}-\frac {x^3 a^2}{b}}{(c+d x)^2 \sqrt {b x^2+a}}dx}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {\frac {c^2 \left (b^2 c^4-6 a b d^2 c^2-2 a^2 d^4\right ) a^3}{b d \left (b c^2+a d^2\right )^2}+\left (\frac {c^2}{d}+\frac {a d}{b}\right ) x^2 a^2-c \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) x a^2}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\int \frac {a^2 c \left (\frac {a c d \left (b^2 c^4-6 a b d^2 c^2-2 a^2 d^4\right )}{\left (b c^2+a d^2\right )^2}-2 \left (b c^2+a d^2\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {a^2 c \int \frac {\frac {a c d \left (b^2 c^4-6 a b d^2 c^2-2 a^2 d^4\right )}{\left (b c^2+a d^2\right )^2}-2 \left (b c^2+a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {a^2 c \left (\frac {b^2 c^5 \left (7 a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )^2}-\frac {2 \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {a^2 c \left (\frac {b^2 c^5 \left (7 a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )^2}-\frac {2 \left (a d^2+b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {a^2 c \left (\frac {b^2 c^5 \left (7 a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )^2}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {a^2 c \left (-\frac {b^2 c^5 \left (7 a d^2+2 b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d \left (a d^2+b c^2\right )^2}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}+\frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a^3 \left (-a d^2+b c^2-2 b c d x\right )}{3 b^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}+\frac {-\frac {3 \left (-\frac {\frac {a^2 c \left (-\frac {b^2 c^5 \left (7 a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \left (a d^2+b c^2\right )^{5/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b d^2}+\frac {a^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a^2 b c^7 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^3 \left (3 \left (-2 a^2 d^4-3 a b c^2 d^2+3 b^2 c^4\right )-4 b c d x \left (2 a d^2+5 b c^2\right )\right )}{b^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}}{3 a}\)

Input:

Int[x^7/((c + d*x)^2*(a + b*x^2)^(5/2)),x]
 

Output:

(a^3*(b*c^2 - a*d^2 - 2*b*c*d*x))/(3*b^3*(b*c^2 + a*d^2)^2*(a + b*x^2)^(3/ 
2)) + (-((a^3*(3*(3*b^2*c^4 - 3*a*b*c^2*d^2 - 2*a^2*d^4) - 4*b*c*d*(5*b*c^ 
2 + 2*a*d^2)*x))/(b^3*(b*c^2 + a*d^2)^3*Sqrt[a + b*x^2])) - (3*(-((a^2*b*c 
^7*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)^3*(c + d*x))) - ((a^2*(b*c^2 + a* 
d^2)*Sqrt[a + b*x^2])/(b^2*d^2) + (a^2*c*((-2*(b*c^2 + a*d^2)*ArcTanh[(Sqr 
t[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (b^2*c^5*(2*b*c^2 + 7*a*d^2)*ArcTa 
nh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*(b*c^2 + a*d^2 
)^(5/2))))/(b*d^2))/(b*c^2 + a*d^2)))/(a*b))/(3*a)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 488

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

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 719

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

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1792\) vs. \(2(268)=536\).

Time = 0.51 (sec) , antiderivative size = 1793, normalized size of antiderivative = 6.08

method	result	size
default	\(\text {Expression too large to display}\)	\(1793\)
risch	\(\text {Expression too large to display}\)	\(2513\)

Input:

int(x^7/(d*x+c)^2/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/d^2*(x^4/b/(b*x^2+a)^(3/2)-4*a/b*(-x^2/b/(b*x^2+a)^(3/2)-2/3*a/b^2/(b*x^ 
2+a)^(3/2)))-6*c^5/d^7*(1/3*x/a/(b*x^2+a)^(3/2)+2/3/a^2/(b*x^2+a)^(1/2)*x) 
-2*c/d^3*(-1/3*x^3/b/(b*x^2+a)^(3/2)+1/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*l 
n(b^(1/2)*x+(b*x^2+a)^(1/2))))+3*c^2/d^4*(-x^2/b/(b*x^2+a)^(3/2)-2/3*a/b^2 
/(b*x^2+a)^(3/2))-4*c^3/d^5*(-1/2*x/b/(b*x^2+a)^(3/2)+1/2*a/b*(1/3*x/a/(b* 
x^2+a)^(3/2)+2/3/a^2/(b*x^2+a)^(1/2)*x))-5/3*c^4/d^6/b/(b*x^2+a)^(3/2)+7/d 
^8*c^6*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d 
^2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2 
)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2) 
+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*( 
x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/ 
(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2* 
b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2 
/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2 
)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2* 
((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^ 
(1/2))/(x+c/d))))-c^7/d^9*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)/(b*(x+c/d)^2-2*b*c 
/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+5*b*c*d/(a*d^2+b*c^2)*(1/3/(a*d^2+b*c^ 
2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+b*c*d/(a*d^2+ 
b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^7/(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^{7}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )^{2}}\, dx \] Input:

integrate(x**7/(d*x+c)**2/(b*x**2+a)**(5/2),x)
 

Output:

Integral(x**7/((a + b*x**2)**(5/2)*(c + d*x)**2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (269) = 538\).

Time = 0.32 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.32 \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^7/(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

-5*b^2*c^9*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d^5 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^ 
4*d^7 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^9 + sqrt(b*x^2 + a)*a^4*d^11) - 5/3* 
b^2*c^9*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d^7 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2 
*d^9 + (b*x^2 + a)^(3/2)*a^3*d^11) - 10/3*b^2*c^9*x/(sqrt(b*x^2 + a)*a^2*b 
^2*c^4*d^7 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^9 + sqrt(b*x^2 + a)*a^4*d^11) - 
 5*b*c^8/(sqrt(b*x^2 + a)*b^3*c^6*d^4 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^6 + 
3*sqrt(b*x^2 + a)*a^2*b*c^2*d^8 + sqrt(b*x^2 + a)*a^3*d^10) - 5/3*b*c^8/(( 
b*x^2 + a)^(3/2)*b^2*c^4*d^6 + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^8 + (b*x^2 + 
a)^(3/2)*a^2*d^10) + 7*b*c^7*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^5 + 2*sqrt(b*x 
^2 + a)*a^2*b*c^2*d^7 + sqrt(b*x^2 + a)*a^3*d^9) + 11/3*b*c^7*x/((b*x^2 + 
a)^(3/2)*a*b*c^2*d^7 + (b*x^2 + a)^(3/2)*a^2*d^9) + 22/3*b*c^7*x/(sqrt(b*x 
^2 + a)*a^2*b*c^2*d^7 + sqrt(b*x^2 + a)*a^3*d^9) + c^7/((b*x^2 + a)^(3/2)* 
b*c^2*d^7*x + (b*x^2 + a)^(3/2)*a*d^9*x + (b*x^2 + a)^(3/2)*b*c^3*d^6 + (b 
*x^2 + a)^(3/2)*a*c*d^8) + 7*c^6/(sqrt(b*x^2 + a)*b^2*c^4*d^4 + 2*sqrt(b*x 
^2 + a)*a*b*c^2*d^6 + sqrt(b*x^2 + a)*a^2*d^8) + 7/3*c^6/((b*x^2 + a)^(3/2 
)*b*c^2*d^6 + (b*x^2 + a)^(3/2)*a*d^8) + x^4/((b*x^2 + a)^(3/2)*b*d^2) + 6 
*c*x^3/((b*x^2 + a)^(3/2)*b*d^3) - 4/3*c^4*x^2/(sqrt(b*x^2 + a)*a^2*d^6) - 
 2/3*c^4*x^2/((b*x^2 + a)^(3/2)*a*d^6) - 7*c^2*x^2/((b*x^2 + a)^(3/2)*b*d^ 
4) - 2*c^2*x^2/(sqrt(b*x^2 + a)*a*b*d^4) + 4*a*x^2/((b*x^2 + a)^(3/2)*b^2* 
d^2) - 4*c^5*x/(sqrt(b*x^2 + a)*a^2*d^7) - 2*c^5*x/((b*x^2 + a)^(3/2)*a...
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^7/(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^7}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(x^7/((a + b*x^2)^(5/2)*(c + d*x)^2),x)
 

Output:

int(x^7/((a + b*x^2)^(5/2)*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 4031, normalized size of antiderivative = 13.66 \[ \int \frac {x^7}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^7/(d*x+c)^2/(b*x^2+a)^(5/2),x)
 

Output:

(21*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
 + b*c*x)*a**3*b**3*c**7*d**2 + 21*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x* 
*2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**3*c**6*d**3*x + 6*sqrt(a* 
d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a 
**2*b**4*c**9 + 6*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + 
 b*c**2) - a*d + b*c*x)*a**2*b**4*c**8*d*x + 42*sqrt(a*d**2 + b*c**2)*log( 
sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**4*c**7*d**2* 
x**2 + 42*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) 
 - a*d + b*c*x)*a**2*b**4*c**6*d**3*x**3 + 12*sqrt(a*d**2 + b*c**2)*log(sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**5*c**9*x**2 + 12* 
sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b 
*c*x)*a*b**5*c**8*d*x**3 + 21*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**5*c**7*d**2*x**4 + 21*sqrt(a*d**2 
 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b** 
5*c**6*d**3*x**5 + 6*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d** 
2 + b*c**2) - a*d + b*c*x)*b**6*c**9*x**4 + 6*sqrt(a*d**2 + b*c**2)*log(sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**6*c**8*d*x**5 - 21* 
sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b**3*c**7*d**2 - 21*sqrt(a*d**2 + 
b*c**2)*log(c + d*x)*a**3*b**3*c**6*d**3*x - 6*sqrt(a*d**2 + b*c**2)*log(c 
 + d*x)*a**2*b**4*c**9 - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**4...