\(\int \frac {x^6}{(c+d x)^3 (a+b x^2)^{3/2}} \, dx\) [1247]

Optimal result

Integrand size = 22, antiderivative size = 298 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{b^2 \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^2 d^3}-\frac {c^6 \sqrt {a+b x^2}}{2 d^3 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {c^5 \left (5 b c^2+12 a d^2\right ) \sqrt {a+b x^2}}{2 d^3 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {3 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^4}-\frac {3 c^4 \left (2 b^2 c^4+7 a b c^2 d^2+10 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^4 \left (b c^2+a d^2\right )^{7/2}} \] Output:

-a^2*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/b^2/(a*d^2+b*c^2)^3/(b* 
x^2+a)^(1/2)+(b*x^2+a)^(1/2)/b^2/d^3-1/2*c^6*(b*x^2+a)^(1/2)/d^3/(a*d^2+b* 
c^2)^2/(d*x+c)^2+1/2*c^5*(12*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/d^3/(a*d^2+b*c 
^2)^3/(d*x+c)-3*c*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)/d^4-3/2*c^4*( 
10*a^2*d^4+7*a*b*c^2*d^2+2*b^2*c^4)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/ 
2)/(b*x^2+a)^(1/2))/d^4/(a*d^2+b*c^2)^(7/2)
 

Mathematica [A] (verified)

Time = 10.92 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.11 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\sqrt {a+b x^2} \left (\frac {1}{b^2 d^3}-\frac {c^6}{2 d^3 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {5 b c^7+12 a c^5 d^2}{2 \left (b c^2 d+a d^3\right )^3 (c+d x)}+\frac {a^2 \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}\right )+\frac {3 c^4 \left (2 b^2 c^4+7 a b c^2 d^2+10 a^2 d^4\right ) \log (c+d x)}{2 d^4 \left (b c^2+a d^2\right )^{7/2}}-\frac {3 c \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{b^{3/2} d^4}-\frac {3 c^4 \left (2 b^2 c^4+7 a b c^2 d^2+10 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{2 d^4 \left (b c^2+a d^2\right )^{7/2}} \] Input:

Integrate[x^6/((c + d*x)^3*(a + b*x^2)^(3/2)),x]
 

Output:

Sqrt[a + b*x^2]*(1/(b^2*d^3) - c^6/(2*d^3*(b*c^2 + a*d^2)^2*(c + d*x)^2) + 
 (5*b*c^7 + 12*a*c^5*d^2)/(2*(b*c^2*d + a*d^3)^3*(c + d*x)) + (a^2*(a^2*d^ 
3 - b^2*c^3*x - 3*a*b*c*d*(c - d*x)))/(b^2*(b*c^2 + a*d^2)^3*(a + b*x^2))) 
 + (3*c^4*(2*b^2*c^4 + 7*a*b*c^2*d^2 + 10*a^2*d^4)*Log[c + d*x])/(2*d^4*(b 
*c^2 + a*d^2)^(7/2)) - (3*c*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(b^(3/2)*d 
^4) - (3*c^4*(2*b^2*c^4 + 7*a*b*c^2*d^2 + 10*a^2*d^4)*Log[a*d - b*c*x + Sq 
rt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(2*d^4*(b*c^2 + a*d^2)^(7/2))
 

Rubi [A] (verified)

Time = 3.42 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {601, 25, 2182, 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.

\(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/2} (c+d x)^3} \, dx\)

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int -\frac {-\frac {8 c^3 d^3 x a^4}{b \left (b c^2+a d^2\right )^3}+\frac {c^4 \left (b c^2-3 a d^2\right ) a^3}{b \left (b c^2+a d^2\right )^3}-\frac {c^2 \left (b^2 c^4+3 a b d^2 c^2+6 a^2 d^4\right ) x^2 a^2}{b \left (b c^2+a d^2\right )^3}+\frac {x^4 a}{b}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-\frac {8 c^3 d^3 x a^4}{b \left (b c^2+a d^2\right )^3}+\frac {c^4 \left (b c^2-3 a d^2\right ) a^3}{b \left (b c^2+a d^2\right )^3}-\frac {c^2 \left (b^2 c^4+3 a b d^2 c^2+6 a^2 d^4\right ) x^2 a^2}{b \left (b c^2+a d^2\right )^3}+\frac {x^4 a}{b}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {\int \frac {\frac {2 a^2 \left (b^2 c^4+3 a^2 d^4\right ) c^3}{b d^2 \left (b c^2+a d^2\right )^2}-\frac {a \left (b^3 c^6+3 a b^2 d^2 c^4-10 a^3 d^6\right ) x c^2}{b d^3 \left (b c^2+a d^2\right )^2}+2 a \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) x^2 c-2 a \left (\frac {c^2}{d}+\frac {a d}{b}\right ) x^3}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\frac {3 a^2 \left (b^2 c^4+4 a b d^2 c^2-2 a^2 d^4\right ) c^2}{b d^2 \left (b c^2+a d^2\right )}-\frac {4 a \left (b c^2+a d^2\right )^2 x c}{b d^3}+\frac {2 a \left (b c^2+a d^2\right )^2 x^2}{b d^2}}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {3 a c \left (\frac {a c d \left (b^2 c^4+4 a b d^2 c^2-2 a^2 d^4\right )}{b c^2+a d^2}-2 \left (b c^2+a d^2\right )^2 x\right )}{d (c+d x) \sqrt {b x^2+a}}dx}{b d^2}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \int \frac {\frac {a c d \left (b^2 c^4+4 a b d^2 c^2-2 a^2 d^4\right )}{b c^2+a d^2}-2 \left (b c^2+a d^2\right )^2 x}{(c+d x) \sqrt {b x^2+a}}dx}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \left (\frac {b c^3 \left (10 a^2 d^4+7 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {2 \left (a d^2+b c^2\right )^2 \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \left (\frac {b c^3 \left (10 a^2 d^4+7 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {2 \left (a d^2+b c^2\right )^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \left (\frac {b c^3 \left (10 a^2 d^4+7 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}\right )}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \left (-\frac {b c^3 \left (10 a^2 d^4+7 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d \left (a d^2+b c^2\right )}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}\right )}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {\frac {3 a c \left (-\frac {b c^3 \left (10 a^2 d^4+7 a b c^2 d^2+2 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \left (a d^2+b c^2\right )^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}\right )}{b d^3}+\frac {2 a \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{b^2 d^3}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2} \left (12 a d^2+5 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{2 \left (a d^2+b c^2\right )}-\frac {a c^6 \sqrt {a+b x^2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^3}\)

Input:

Int[x^6/((c + d*x)^3*(a + b*x^2)^(3/2)),x]
 

Output:

-((a^2*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(b^2*(b*c^2 + a* 
d^2)^3*Sqrt[a + b*x^2])) + (-1/2*(a*c^6*Sqrt[a + b*x^2])/(d^3*(b*c^2 + a*d 
^2)^2*(c + d*x)^2) - (-((a*c^5*(5*b*c^2 + 12*a*d^2)*Sqrt[a + b*x^2])/(d^3* 
(b*c^2 + a*d^2)^2*(c + d*x))) - ((2*a*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2])/( 
b^2*d^3) + (3*a*c*((-2*(b*c^2 + a*d^2)^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^ 
2]])/(Sqrt[b]*d) - (b*c^3*(2*b^2*c^4 + 7*a*b*c^2*d^2 + 10*a^2*d^4)*ArcTanh 
[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*(b*c^2 + a*d^2)^ 
(3/2))))/(b*d^3))/(b*c^2 + a*d^2))/(2*(b*c^2 + a*d^2)))/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 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. \(1162\) vs. \(2(272)=544\).

Time = 0.56 (sec) , antiderivative size = 1163, normalized size of antiderivative = 3.90

method	result	size
risch	\(\text {Expression too large to display}\)	\(1163\)
default	\(\text {Expression too large to display}\)	\(1892\)

Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(b*x^2+a)^(1/2)/b^2/d^3-1/b/d^3*(1/d^4*b^2*c^6/((-a*b)^(1/2)*d+b*c)/((-a*b 
)^(1/2)*d-b*c)*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*(x+c 
/d)+(a*d^2+b*c^2)/d^2)^(1/2)+3/2*b*c*d/(a*d^2+b*c^2)*(-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)^(1/2)-b*c*d/(a*d^ 
2+b*c^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)))+1/2*b/(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)))+3*c/d*ln(b^(1/2)*x+ 
(b*x^2+a)^(1/2))/b^(1/2)+1/2*d^3*a^2/((-a*b)^(1/2)*d+b*c)^3/(x-(-a*b)^(1/2 
)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)-1/2* 
d^3*a^2/((-a*b)^(1/2)*d-b*c)^3/(x+(-a*b)^(1/2)/b)*((x+(-a*b)^(1/2)/b)^2*b- 
2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)+1/d^2*b^3*c^4*(6*(-a*b)^(3/2)*c*d 
^3+6*(-a*b)^(1/2)*a*b*c*d^3-15*a^2*b*d^4-17*a*b^2*c^2*d^2-6*b^3*c^4)/((-a* 
b)^(1/2)*d+b*c)^3/((-a*b)^(1/2)*d-b*c)^3/((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))+2/d^3*b^3*c^5*(3*a*d^2+ 
2*b*c^2)/((-a*b)^(1/2)*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2*(-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)^(1/2)-b*c*d/(a* 
d^2+b*c^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...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate(x**6/(d*x+c)**3/(b*x**2+a)**(3/2),x)
 

Output:

Integral(x**6/((a + b*x**2)**(3/2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (273) = 546\).

Time = 0.13 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.33 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

15/2*b^3*c^9*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d^6 + 3*sqrt(b*x^2 + a)*a^2*b^2* 
c^4*d^8 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^10 + sqrt(b*x^2 + a)*a^4*d^12) + 1 
5/2*b^2*c^8/(sqrt(b*x^2 + a)*b^3*c^6*d^5 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^7 
 + 3*sqrt(b*x^2 + a)*a^2*b*c^2*d^9 + sqrt(b*x^2 + a)*a^3*d^11) - 49/2*b^2* 
c^7*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^6 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^8 + s 
qrt(b*x^2 + a)*a^3*d^10) - 5/2*b*c^7/(sqrt(b*x^2 + a)*b^2*c^4*d^6*x + 2*sq 
rt(b*x^2 + a)*a*b*c^2*d^8*x + sqrt(b*x^2 + a)*a^2*d^10*x + sqrt(b*x^2 + a) 
*b^2*c^5*d^5 + 2*sqrt(b*x^2 + a)*a*b*c^3*d^7 + sqrt(b*x^2 + a)*a^2*c*d^9) 
- 39/2*b*c^6/(sqrt(b*x^2 + a)*b^2*c^4*d^5 + 2*sqrt(b*x^2 + a)*a*b*c^2*d^7 
+ sqrt(b*x^2 + a)*a^2*d^9) + 27*b*c^5*x/(sqrt(b*x^2 + a)*a*b*c^2*d^6 + sqr 
t(b*x^2 + a)*a^2*d^8) - 1/2*c^6/(sqrt(b*x^2 + a)*b*c^2*d^7*x^2 + sqrt(b*x^ 
2 + a)*a*d^9*x^2 + 2*sqrt(b*x^2 + a)*b*c^3*d^6*x + 2*sqrt(b*x^2 + a)*a*c*d 
^8*x + sqrt(b*x^2 + a)*b*c^4*d^5 + sqrt(b*x^2 + a)*a*c^2*d^7) + 6*c^5/(sqr 
t(b*x^2 + a)*b*c^2*d^6*x + sqrt(b*x^2 + a)*a*d^8*x + sqrt(b*x^2 + a)*b*c^3 
*d^5 + sqrt(b*x^2 + a)*a*c*d^7) + 15*c^4/(sqrt(b*x^2 + a)*b*c^2*d^5 + sqrt 
(b*x^2 + a)*a*d^7) + x^2/(sqrt(b*x^2 + a)*b*d^3) - 10*c^3*x/(sqrt(b*x^2 + 
a)*a*d^6) + 3*c*x/(sqrt(b*x^2 + a)*b*d^4) - 3*c*arcsinh(b*x/sqrt(a*b))/(b^ 
(3/2)*d^4) + 15/2*b^2*c^8*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sq 
rt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(7/2)*d^11) - 39/2*b*c^6*arcsinh(b 
*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1016 vs. \(2 (273) = 546\).

Time = 0.20 (sec) , antiderivative size = 1016, normalized size of antiderivative = 3.41 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

(((b^8*c^12*d^11 + 6*a*b^7*c^10*d^13 + 15*a^2*b^6*c^8*d^15 + 20*a^3*b^5*c^ 
6*d^17 + 15*a^4*b^4*c^4*d^19 + 6*a^5*b^3*c^2*d^21 + a^6*b^2*d^23)*x/(b^9*c 
^12*d^14 + 6*a*b^8*c^10*d^16 + 15*a^2*b^7*c^8*d^18 + 20*a^3*b^6*c^6*d^20 + 
 15*a^4*b^5*c^4*d^22 + 6*a^5*b^4*c^2*d^24 + a^6*b^3*d^26) - (a^2*b^6*c^9*d 
^14 - 6*a^4*b^4*c^5*d^18 - 8*a^5*b^3*c^3*d^20 - 3*a^6*b^2*c*d^22)/(b^9*c^1 
2*d^14 + 6*a*b^8*c^10*d^16 + 15*a^2*b^7*c^8*d^18 + 20*a^3*b^6*c^6*d^20 + 1 
5*a^4*b^5*c^4*d^22 + 6*a^5*b^4*c^2*d^24 + a^6*b^3*d^26))*x + (a*b^7*c^12*d 
^11 + 6*a^2*b^6*c^10*d^13 + 12*a^3*b^5*c^8*d^15 + 12*a^4*b^4*c^6*d^17 + 9* 
a^5*b^3*c^4*d^19 + 6*a^6*b^2*c^2*d^21 + 2*a^7*b*d^23)/(b^9*c^12*d^14 + 6*a 
*b^8*c^10*d^16 + 15*a^2*b^7*c^8*d^18 + 20*a^3*b^6*c^6*d^20 + 15*a^4*b^5*c^ 
4*d^22 + 6*a^5*b^4*c^2*d^24 + a^6*b^3*d^26))/sqrt(b*x^2 + a) - 3*(2*b^2*c^ 
8 + 7*a*b*c^6*d^2 + 10*a^2*c^4*d^4)*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))* 
d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^3*c^6*d^4 + 3*a*b^2*c^4*d^6 + 3*a 
^2*b*c^2*d^8 + a^3*d^10)*sqrt(-b*c^2 - a*d^2)) + (6*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^3*b^2*c^8*d + 13*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c^6*d^3 + 10* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^9 + 19*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*a*b^(3/2)*c^7*d^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)* 
c^5*d^4 - 14*(sqrt(b)*x - sqrt(b*x^2 + a))*a*b^2*c^8*d - 35*(sqrt(b)*x - s 
qrt(b*x^2 + a))*a^2*b*c^6*d^3 + 5*a^2*b^(3/2)*c^7*d^2 + 12*a^3*sqrt(b)*c^5 
*d^4)/((b^3*c^6*d^4 + 3*a*b^2*c^4*d^6 + 3*a^2*b*c^2*d^8 + a^3*d^10)*((s...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int(x^6/((a + b*x^2)^(3/2)*(c + d*x)^3),x)
 

Output:

int(x^6/((a + b*x^2)^(3/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 3958, normalized size of antiderivative = 13.28 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^(3/2),x)
 

Output:

(30*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**2*c**6*d**4 + 60*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**2*c**5*d**5*x + 30*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**2*c**4*d**6*x**2 + 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**2*b**3*c**8*d**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* 
*3*c**7*d**3*x + 51*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**3*c**6*d**4*x**2 + 60*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**3*c* 
*5*d**5*x**3 + 30*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**3*c**4*d**6*x**4 + 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*b**4*c**10 + 
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**4*c**9*d*x + 27*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**4*c**8*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*b** 
4*c**7*d**3*x**3 + 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**4*c**6*d**4*x**4 + 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**5*c**10*...