\(\int \frac {x^2 (a+b x^2)^p}{(c+d x)^3} \, dx\) [1859]

Optimal result

Integrand size = 20, antiderivative size = 396 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(c+d x)^3} \, dx=-\frac {c^2 \left (a+b x^2\right )^{1+p}}{2 d \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {c \left (2 a d^2+b c^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{d \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\left (a^2 d^4+a b c^2 d^2 (2+5 p)+b^2 c^4 \left (1+3 p+2 p^2\right )\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c d^2 \left (b c^2+a d^2\right )^2}-\frac {b c (1+2 p) \left (2 a d^2+b c^2 (1+p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d^2 \left (b c^2+a d^2\right )^2}-\frac {\left (a^2 d^4+a b c^2 d^2 (2+5 p)+b^2 c^4 \left (1+3 p+2 p^2\right )\right ) \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {d^2 \left (a+b x^2\right )}{b c^2+a d^2}\right )}{2 d \left (b c^2+a d^2\right )^3 (1+p)} \] Output:

-1/2*c^2*(b*x^2+a)^(p+1)/d/(a*d^2+b*c^2)/(d*x+c)^2+c*(2*a*d^2+b*c^2*(p+1)) 
*(b*x^2+a)^(p+1)/d/(a*d^2+b*c^2)^2/(d*x+c)+(a^2*d^4+a*b*c^2*d^2*(2+5*p)+b^ 
2*c^4*(2*p^2+3*p+1))*x*(b*x^2+a)^p*AppellF1(1/2,1,-p,3/2,d^2*x^2/c^2,-b*x^ 
2/a)/c/d^2/(a*d^2+b*c^2)^2/((1+b*x^2/a)^p)-b*c*(1+2*p)*(2*a*d^2+b*c^2*(p+1 
))*x*(b*x^2+a)^p*hypergeom([1/2, -p],[3/2],-b*x^2/a)/d^2/(a*d^2+b*c^2)^2/( 
(1+b*x^2/a)^p)-1/2*(a^2*d^4+a*b*c^2*d^2*(2+5*p)+b^2*c^4*(2*p^2+3*p+1))*(b* 
x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],d^2*(b*x^2+a)/(a*d^2+b*c^2))/d/(a*d^ 
2+b*c^2)^3/(p+1)
 

Mathematica [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(c+d x)^3} \, dx=\frac {\left (\frac {d \left (-\sqrt {-\frac {a}{b}}+x\right )}{c+d x}\right )^{-p} \left (\frac {d \left (\sqrt {-\frac {a}{b}}+x\right )}{c+d x}\right )^{-p} \left (a+b x^2\right )^p \left (-\frac {4 c \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {c-\sqrt {-\frac {a}{b}} d}{c+d x},\frac {c+\sqrt {-\frac {a}{b}} d}{c+d x}\right )}{(-1+2 p) (c+d x)}+\frac {c^2 \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {c-\sqrt {-\frac {a}{b}} d}{c+d x},\frac {c+\sqrt {-\frac {a}{b}} d}{c+d x}\right )}{(-1+p) (c+d x)^2}+\frac {\operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c-\sqrt {-\frac {a}{b}} d}{c+d x},\frac {c+\sqrt {-\frac {a}{b}} d}{c+d x}\right )}{p}\right )}{2 d^3} \] Input:

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

Output:

((a + b*x^2)^p*((-4*c*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (c - Sqrt[-(a/b)] 
*d)/(c + d*x), (c + Sqrt[-(a/b)]*d)/(c + d*x)])/((-1 + 2*p)*(c + d*x)) + ( 
c^2*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (c - Sqrt[-(a/b)]*d)/(c + d*x), (c 
+ Sqrt[-(a/b)]*d)/(c + d*x)])/((-1 + p)*(c + d*x)^2) + AppellF1[-2*p, -p, 
-p, 1 - 2*p, (c - Sqrt[-(a/b)]*d)/(c + d*x), (c + Sqrt[-(a/b)]*d)/(c + d*x 
)]/p))/(2*d^3*((d*(-Sqrt[-(a/b)] + x))/(c + d*x))^p*((d*(Sqrt[-(a/b)] + x) 
)/(c + d*x))^p)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {603, 27, 688, 719, 238, 237, 504, 334, 333, 353, 78}

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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 238

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

\(\Big \downarrow \) 237

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \int \frac {\left (b x^2+a\right )^p}{c+d x}dx}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 504

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \left (c \int \frac {\left (b x^2+a\right )^p}{c^2-d^2 x^2}dx-d \int \frac {x \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\right )}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 334

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \left (c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {\left (\frac {b x^2}{a}+1\right )^p}{c^2-d^2 x^2}dx-d \int \frac {x \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\right )}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 333

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c}-d \int \frac {x \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\right )}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 353

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c}-\frac {1}{2} d \int \frac {\left (b x^2+a\right )^p}{c^2-d^2 x^2}dx^2\right )}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 78

\(\displaystyle -\frac {-\frac {\frac {\left (a^2 d^4+a b c^2 d^2 (5 p+2)+b^2 c^4 \left (2 p^2+3 p+1\right )\right ) \left (\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c}-\frac {d \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {d^2 \left (b x^2+a\right )}{b c^2+a d^2}\right )}{2 (p+1) \left (a d^2+b c^2\right )}\right )}{d}-\frac {b c (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a d^2+b c^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{d}}{a d^2+b c^2}-\frac {c \left (a+b x^2\right )^{p+1} \left (2 a d^2+b c^2 (p+1)\right )}{(c+d x) \left (a d^2+b c^2\right )}}{d \left (a d^2+b c^2\right )}-\frac {c^2 \left (a+b x^2\right )^{p+1}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/2*(c^2*(a + b*x^2)^(1 + p))/(d*(b*c^2 + a*d^2)*(c + d*x)^2) - (-((c*(2* 
a*d^2 + b*c^2*(1 + p))*(a + b*x^2)^(1 + p))/((b*c^2 + a*d^2)*(c + d*x))) - 
 (-((b*c*(1 + 2*p)*(2*a*d^2 + b*c^2*(1 + p))*x*(a + b*x^2)^p*Hypergeometri 
c2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(d*(1 + (b*x^2)/a)^p)) + ((a^2*d^4 + a*b 
*c^2*d^2*(2 + 5*p) + b^2*c^4*(1 + 3*p + 2*p^2))*((x*(a + b*x^2)^p*AppellF1 
[1/2, -p, 1, 3/2, -((b*x^2)/a), (d^2*x^2)/c^2])/(c*(1 + (b*x^2)/a)^p) - (d 
*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (d^2*(a + b*x^2))/ 
(b*c^2 + a*d^2)])/(2*(b*c^2 + a*d^2)*(1 + p))))/d)/(b*c^2 + a*d^2))/(d*(b* 
c^2 + a*d^2))
 

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 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^2 + a)^p*x^2/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*(a + b*x**2)**p*a*d - 2*(a + b*x**2)**p*b*c*p*x - 2*(a + b*x**2)**p* 
b*c*x + 2*(a + b*x**2)**p*b*d*p*x**2 - (a + b*x**2)**p*b*d*x**2 - 12*int(( 
a + b*x**2)**p/(2*a*c**3*p - a*c**3 + 6*a*c**2*d*p*x - 3*a*c**2*d*x + 6*a* 
c*d**2*p*x**2 - 3*a*c*d**2*x**2 + 2*a*d**3*p*x**3 - a*d**3*x**3 + 2*b*c**3 
*p*x**2 - b*c**3*x**2 + 6*b*c**2*d*p*x**3 - 3*b*c**2*d*x**3 + 6*b*c*d**2*p 
*x**4 - 3*b*c*d**2*x**4 + 2*b*d**3*p*x**5 - b*d**3*x**5),x)*a**2*c**2*d**2 
*p + 6*int((a + b*x**2)**p/(2*a*c**3*p - a*c**3 + 6*a*c**2*d*p*x - 3*a*c** 
2*d*x + 6*a*c*d**2*p*x**2 - 3*a*c*d**2*x**2 + 2*a*d**3*p*x**3 - a*d**3*x** 
3 + 2*b*c**3*p*x**2 - b*c**3*x**2 + 6*b*c**2*d*p*x**3 - 3*b*c**2*d*x**3 + 
6*b*c*d**2*p*x**4 - 3*b*c*d**2*x**4 + 2*b*d**3*p*x**5 - b*d**3*x**5),x)*a* 
*2*c**2*d**2 - 24*int((a + b*x**2)**p/(2*a*c**3*p - a*c**3 + 6*a*c**2*d*p* 
x - 3*a*c**2*d*x + 6*a*c*d**2*p*x**2 - 3*a*c*d**2*x**2 + 2*a*d**3*p*x**3 - 
 a*d**3*x**3 + 2*b*c**3*p*x**2 - b*c**3*x**2 + 6*b*c**2*d*p*x**3 - 3*b*c** 
2*d*x**3 + 6*b*c*d**2*p*x**4 - 3*b*c*d**2*x**4 + 2*b*d**3*p*x**5 - b*d**3* 
x**5),x)*a**2*c*d**3*p*x + 12*int((a + b*x**2)**p/(2*a*c**3*p - a*c**3 + 6 
*a*c**2*d*p*x - 3*a*c**2*d*x + 6*a*c*d**2*p*x**2 - 3*a*c*d**2*x**2 + 2*a*d 
**3*p*x**3 - a*d**3*x**3 + 2*b*c**3*p*x**2 - b*c**3*x**2 + 6*b*c**2*d*p*x* 
*3 - 3*b*c**2*d*x**3 + 6*b*c*d**2*p*x**4 - 3*b*c*d**2*x**4 + 2*b*d**3*p*x* 
*5 - b*d**3*x**5),x)*a**2*c*d**3*x - 12*int((a + b*x**2)**p/(2*a*c**3*p - 
a*c**3 + 6*a*c**2*d*p*x - 3*a*c**2*d*x + 6*a*c*d**2*p*x**2 - 3*a*c*d**2...