\(\int \frac {x^6}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1297]

Optimal result

Integrand size = 22, antiderivative size = 328 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {a^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{3 b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}+\frac {a \left (3 a d \left (9 b c^4-4 a c^2 d^2-\frac {a^2 d^4}{b}\right )+c \left (7 b^2 c^4-32 a b c^2 d^2-3 a^2 d^4\right ) x\right )}{3 b \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}-\frac {c^6 \sqrt {a+b x^2}}{2 d \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {c^5 \left (b c^2+12 a d^2\right ) \sqrt {a+b x^2}}{2 d \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {5 a c^4 \left (b c^2-6 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

-1/3*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)^(3/2)+1/3*a*(3*a*d*(9*b*c^4-4*a*c^2*d^2-a^2*d^4/b)+c*(-3*a^2*d^ 
4-32*a*b*c^2*d^2+7*b^2*c^4)*x)/b/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)-1/2*c^6*( 
b*x^2+a)^(1/2)/d/(a*d^2+b*c^2)^3/(d*x+c)^2+1/2*c^5*(12*a*d^2+b*c^2)*(b*x^2 
+a)^(1/2)/d/(a*d^2+b*c^2)^4/(d*x+c)+5/2*a*c^4*(-6*a*d^2+b*c^2)*arctanh((-b 
*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2)^(9/2)
 

Mathematica [A] (verified)

Time = 10.78 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.07 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {3 b^5 c^7 x^5-4 a^5 d^5 (c+d x)^2-2 a^4 b d^3 (c+d x)^2 \left (14 c^2+3 d^2 x^2\right )+a b^4 c^5 x^3 \left (20 c^2+61 c d x+50 d^2 x^2\right )+a^2 b^3 c^3 x \left (15 c^4+144 c^3 d x+128 c^2 d^2 x^2-74 c d^3 x^3-64 d^4 x^4\right )+3 a^3 b^2 c d \left (27 c^5+24 c^4 d x-32 c^3 d^2 x^2-38 c^2 d^3 x^3-12 c d^4 x^4-2 d^5 x^5\right )}{b^2 \left (b c^2+a d^2\right )^4 (c+d x)^2 \left (a+b x^2\right )^{3/2}}-\frac {15 a c^4 \left (b c^2-6 a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}+\frac {15 a c^4 \left (b c^2-6 a d^2\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

Integrate[x^6/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

((3*b^5*c^7*x^5 - 4*a^5*d^5*(c + d*x)^2 - 2*a^4*b*d^3*(c + d*x)^2*(14*c^2 
+ 3*d^2*x^2) + a*b^4*c^5*x^3*(20*c^2 + 61*c*d*x + 50*d^2*x^2) + a^2*b^3*c^ 
3*x*(15*c^4 + 144*c^3*d*x + 128*c^2*d^2*x^2 - 74*c*d^3*x^3 - 64*d^4*x^4) + 
 3*a^3*b^2*c*d*(27*c^5 + 24*c^4*d*x - 32*c^3*d^2*x^2 - 38*c^2*d^3*x^3 - 12 
*c*d^4*x^4 - 2*d^5*x^5))/(b^2*(b*c^2 + a*d^2)^4*(c + d*x)^2*(a + b*x^2)^(3 
/2)) - (15*a*c^4*(b*c^2 - 6*a*d^2)*Log[c + d*x])/(b*c^2 + a*d^2)^(9/2) + ( 
15*a*c^4*(b*c^2 - 6*a*d^2)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + 
b*x^2]])/(b*c^2 + a*d^2)^(9/2))/6
 

Rubi [A] (verified)

Time = 3.52 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {601, 25, 2178, 27, 2182, 25, 27, 679, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {\int \frac {3 \left (\frac {2 a^3 b \left (b c^2-5 a d^2\right ) c^6}{\left (b c^2+a d^2\right )^4}-\frac {24 a^4 b d^3 x c^5}{\left (b c^2+a d^2\right )^4}-\frac {a^2 b \left (b^2 c^4+4 a b d^2 c^2+15 a^2 d^4\right ) x^2 c^4}{\left (b c^2+a d^2\right )^4}\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \int \frac {\frac {2 a^3 b \left (b c^2-5 a d^2\right ) c^6}{\left (b c^2+a d^2\right )^4}-\frac {24 a^4 b d^3 x c^5}{\left (b c^2+a d^2\right )^4}-\frac {a^2 b \left (b^2 c^4+4 a b d^2 c^2+15 a^2 d^4\right ) x^2 c^4}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}-\frac {\int -\frac {a^2 b c^4 \left (6 a c d \left (b c^2-3 a d^2\right )-\left (b^2 c^4+7 a b d^2 c^2+30 a^2 d^4\right ) x\right )}{d \left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {\int \frac {a^2 b c^4 \left (6 a c d \left (b c^2-3 a d^2\right )-\left (b^2 c^4+7 a b d^2 c^2+30 a^2 d^4\right ) x\right )}{d \left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}+\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^4 \int \frac {6 a c d \left (b c^2-3 a d^2\right )-\left (b^2 c^4+7 a b d^2 c^2+30 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 d \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^4 \left (5 a d \left (b c^2-6 a d^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx-\frac {c \sqrt {a+b x^2} \left (12 a d^2+b c^2\right )}{c+d x}\right )}{2 d \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^4 \left (-5 a d \left (b c^2-6 a d^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}-\frac {c \sqrt {a+b x^2} \left (12 a d^2+b c^2\right )}{c+d x}\right )}{2 d \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {a^2 \left (b c x \left (-3 a^2 d^4-32 a b c^2 d^2+7 b^2 c^4\right )+3 a d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right )\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^4 \left (-\frac {5 a d \left (b c^2-6 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a d^2+b c^2}}-\frac {c \sqrt {a+b x^2} \left (12 a d^2+b c^2\right )}{c+d x}\right )}{2 d \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^6 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

Input:

Int[x^6/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

-1/3*(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*(a + b*x^2)^(3/2)) + ((a^2*(3*a*d*(9*b^2*c^4 - 4*a*b*c^2*d^2 - a 
^2*d^4) + b*c*(7*b^2*c^4 - 32*a*b*c^2*d^2 - 3*a^2*d^4)*x))/(b^2*(b*c^2 + a 
*d^2)^4*Sqrt[a + b*x^2]) - (3*((a^2*b*c^6*Sqrt[a + b*x^2])/(2*d*(b*c^2 + a 
*d^2)^3*(c + d*x)^2) + (a^2*b*c^4*(-((c*(b*c^2 + 12*a*d^2)*Sqrt[a + b*x^2] 
)/(c + d*x)) - (5*a*d*(b*c^2 - 6*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 
+ a*d^2]*Sqrt[a + b*x^2])])/Sqrt[b*c^2 + a*d^2]))/(2*d*(b*c^2 + a*d^2)^4)) 
)/(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 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 679

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 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3267\) vs. \(2(304)=608\).

Time = 0.48 (sec) , antiderivative size = 3268, normalized size of antiderivative = 9.96

method	result	size
default	\(\text {Expression too large to display}\)	\(3268\)

Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1294 vs. \(2 (305) = 610\).

Time = 1.00 (sec) , antiderivative size = 2615, normalized size of antiderivative = 7.97 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*(15*(a^3*b^3*c^8 - 6*a^4*b^2*c^6*d^2 + (a*b^5*c^6*d^2 - 6*a^2*b^4*c 
^4*d^4)*x^6 + 2*(a*b^5*c^7*d - 6*a^2*b^4*c^5*d^3)*x^5 + (a*b^5*c^8 - 4*a^2 
*b^4*c^6*d^2 - 12*a^3*b^3*c^4*d^4)*x^4 + 4*(a^2*b^4*c^7*d - 6*a^3*b^3*c^5* 
d^3)*x^3 + (2*a^2*b^4*c^8 - 11*a^3*b^3*c^6*d^2 - 6*a^4*b^2*c^4*d^4)*x^2 + 
2*(a^3*b^3*c^7*d - 6*a^4*b^2*c^5*d^3)*x)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c* 
d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d 
^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(81*a^3* 
b^3*c^8*d + 53*a^4*b^2*c^6*d^3 - 32*a^5*b*c^4*d^5 - 4*a^6*c^2*d^7 + (3*b^6 
*c^9 + 53*a*b^5*c^7*d^2 - 14*a^2*b^4*c^5*d^4 - 70*a^3*b^3*c^3*d^6 - 6*a^4* 
b^2*c*d^8)*x^5 + (61*a*b^5*c^8*d - 13*a^2*b^4*c^6*d^3 - 110*a^3*b^3*c^4*d^ 
5 - 42*a^4*b^2*c^2*d^7 - 6*a^5*b*d^9)*x^4 + 2*(10*a*b^5*c^9 + 74*a^2*b^4*c 
^7*d^2 + 7*a^3*b^3*c^5*d^4 - 63*a^4*b^2*c^3*d^6 - 6*a^5*b*c*d^8)*x^3 + 2*( 
72*a^2*b^4*c^8*d + 24*a^3*b^3*c^6*d^3 - 65*a^4*b^2*c^4*d^5 - 19*a^5*b*c^2* 
d^7 - 2*a^6*d^9)*x^2 + (15*a^2*b^4*c^9 + 87*a^3*b^3*c^7*d^2 + 16*a^4*b^2*c 
^5*d^4 - 64*a^5*b*c^3*d^6 - 8*a^6*c*d^8)*x)*sqrt(b*x^2 + a))/(a^2*b^7*c^12 
 + 5*a^3*b^6*c^10*d^2 + 10*a^4*b^5*c^8*d^4 + 10*a^5*b^4*c^6*d^6 + 5*a^6*b^ 
3*c^4*d^8 + a^7*b^2*c^2*d^10 + (b^9*c^10*d^2 + 5*a*b^8*c^8*d^4 + 10*a^2*b^ 
7*c^6*d^6 + 10*a^3*b^6*c^4*d^8 + 5*a^4*b^5*c^2*d^10 + a^5*b^4*d^12)*x^6 + 
2*(b^9*c^11*d + 5*a*b^8*c^9*d^3 + 10*a^2*b^7*c^7*d^5 + 10*a^3*b^6*c^5*d^7 
+ 5*a^4*b^5*c^3*d^9 + a^5*b^4*c*d^11)*x^5 + (b^9*c^12 + 7*a*b^8*c^10*d^...
 

Sympy [F]

\[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate(x**6/(d*x+c)**3/(b*x**2+a)**(5/2),x)
 

Output:

Integral(x**6/((a + b*x**2)**(5/2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1761 vs. \(2 (305) = 610\).

Time = 0.23 (sec) , antiderivative size = 1761, normalized size of antiderivative = 5.37 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

35/2*b^3*c^9*x/(sqrt(b*x^2 + a)*a*b^4*c^8*d^4 + 4*sqrt(b*x^2 + a)*a^2*b^3* 
c^6*d^6 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^8 + 4*sqrt(b*x^2 + a)*a^4*b*c^2* 
d^10 + sqrt(b*x^2 + a)*a^5*d^12) + 35/6*b^3*c^9*x/((b*x^2 + a)^(3/2)*a*b^3 
*c^6*d^6 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^8 + 3*(b*x^2 + a)^(3/2)*a^3*b 
*c^2*d^10 + (b*x^2 + a)^(3/2)*a^4*d^12) + 35/3*b^3*c^9*x/(sqrt(b*x^2 + a)* 
a^2*b^3*c^6*d^6 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^8 + 3*sqrt(b*x^2 + a)*a^ 
4*b*c^2*d^10 + sqrt(b*x^2 + a)*a^5*d^12) + 35/2*b^2*c^8/(sqrt(b*x^2 + a)*b 
^4*c^8*d^3 + 4*sqrt(b*x^2 + a)*a*b^3*c^6*d^5 + 6*sqrt(b*x^2 + a)*a^2*b^2*c 
^4*d^7 + 4*sqrt(b*x^2 + a)*a^3*b*c^2*d^9 + sqrt(b*x^2 + a)*a^4*d^11) + 35/ 
6*b^2*c^8/((b*x^2 + a)^(3/2)*b^3*c^6*d^5 + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d 
^7 + 3*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^9 + (b*x^2 + a)^(3/2)*a^3*d^11) - 65/ 
2*b^2*c^7*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d^4 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4 
*d^6 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^8 + sqrt(b*x^2 + a)*a^4*d^10) - 31/2* 
b^2*c^7*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d^6 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2 
*d^8 + (b*x^2 + a)^(3/2)*a^3*d^10) - 31*b^2*c^7*x/(sqrt(b*x^2 + a)*a^2*b^2 
*c^4*d^6 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^8 + sqrt(b*x^2 + a)*a^4*d^10) - 7 
/2*b*c^7/((b*x^2 + a)^(3/2)*b^2*c^4*d^6*x + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^ 
8*x + (b*x^2 + a)^(3/2)*a^2*d^10*x + (b*x^2 + a)^(3/2)*b^2*c^5*d^5 + 2*(b* 
x^2 + a)^(3/2)*a*b*c^3*d^7 + (b*x^2 + a)^(3/2)*a^2*c*d^9) - 65/2*b*c^6/(sq 
rt(b*x^2 + a)*b^3*c^6*d^3 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^5 + 3*sqrt(b*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (305) = 610\).

Time = 0.28 (sec) , antiderivative size = 2230, normalized size of antiderivative = 6.80 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-5*(a*b*c^6 - 6*a^2*c^4*d^2)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sq 
rt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4 
*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(-b*c^2 - a*d^2)) + 1/3*((((7*a^2*b^ 
16*c^29 + 52*a^3*b^15*c^27*d^2 + 75*a^4*b^14*c^25*d^4 - 608*a^5*b^13*c^23* 
d^6 - 3773*a^6*b^12*c^21*d^8 - 10956*a^7*b^11*c^19*d^10 - 20361*a^8*b^10*c 
^17*d^12 - 26400*a^9*b^9*c^15*d^14 - 24651*a^10*b^8*c^13*d^16 - 16676*a^11 
*b^7*c^11*d^18 - 8063*a^12*b^6*c^9*d^20 - 2688*a^13*b^5*c^7*d^22 - 575*a^1 
4*b^4*c^5*d^24 - 68*a^15*b^3*c^3*d^26 - 3*a^16*b^2*c*d^28)*x/(a*b^18*c^32 
+ 16*a^2*b^17*c^30*d^2 + 120*a^3*b^16*c^28*d^4 + 560*a^4*b^15*c^26*d^6 + 1 
820*a^5*b^14*c^24*d^8 + 4368*a^6*b^13*c^22*d^10 + 8008*a^7*b^12*c^20*d^12 
+ 11440*a^8*b^11*c^18*d^14 + 12870*a^9*b^10*c^16*d^16 + 11440*a^10*b^9*c^1 
4*d^18 + 8008*a^11*b^8*c^12*d^20 + 4368*a^12*b^7*c^10*d^22 + 1820*a^13*b^6 
*c^8*d^24 + 560*a^14*b^5*c^6*d^26 + 120*a^15*b^4*c^4*d^28 + 16*a^16*b^3*c^ 
2*d^30 + a^17*b^2*d^32) + 3*(9*a^3*b^15*c^28*d + 104*a^4*b^14*c^26*d^3 + 5 
45*a^5*b^13*c^24*d^5 + 1704*a^6*b^12*c^22*d^7 + 3509*a^7*b^11*c^20*d^9 + 4 
928*a^8*b^10*c^18*d^11 + 4653*a^9*b^9*c^16*d^13 + 2640*a^10*b^8*c^14*d^15 
+ 363*a^11*b^7*c^12*d^17 - 792*a^12*b^6*c^10*d^19 - 781*a^13*b^5*c^8*d^21 
- 376*a^14*b^4*c^6*d^23 - 105*a^15*b^3*c^4*d^25 - 16*a^16*b^2*c^2*d^27 - a 
^17*b*d^29)/(a*b^18*c^32 + 16*a^2*b^17*c^30*d^2 + 120*a^3*b^16*c^28*d^4 + 
560*a^4*b^15*c^26*d^6 + 1820*a^5*b^14*c^24*d^8 + 4368*a^6*b^13*c^22*d^1...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int(x^6/((a + b*x^2)^(5/2)*(c + d*x)^3),x)
 

Output:

int(x^6/((a + b*x^2)^(5/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 2759, normalized size of antiderivative = 8.41 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^(5/2),x)
 

Output:

(90*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**4*b**2*c**6*d**2 + 180*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**4*b**2*c**5*d**3*x + 90*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**4*b**2*c**4*d**4*x**2 - 15*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**8 - 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**3*c 
**7*d*x + 165*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**2*x**2 + 360*sqrt(a*d**2 + b*c**2)*l 
og(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**3*c**5*d* 
*3*x**3 + 180*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**4*d**4*x**4 - 30*sqrt(a*d**2 + b*c**2)*lo 
g(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**4*c**8*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**4*c**7*d*x**3 + 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**4*c**6*d**2*x**4 + 1 
80*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**5*d**3*x**5 + 90*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**4*d**4*x**6 - 15 
*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d...