\(\int \frac {x^6}{(c+d x) (a+b x^2)^{5/2}} \, dx\) [1276]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=-\frac {a^2 (a d+b c x)}{3 b^3 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}+\frac {a \left (3 a d \left (3 b c^2+2 a d^2\right )+b c \left (7 b c^2+4 a d^2\right ) x\right )}{3 b^3 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^3 d}-\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2} d^2}-\frac {c^6 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.23 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {d \left (8 a^4 d^4+3 b^4 c^4 x^4+a b^3 c^2 x^2 \left (6 c^2+7 c d x+6 d^2 x^2\right )+a^3 b d^2 \left (14 c^2+3 c d x+12 d^2 x^2\right )+a^2 b^2 \left (3 c^4+6 c^3 d x+21 c^2 d^2 x^2+4 c d^3 x^3+3 d^4 x^4\right )\right )}{b^3 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}-\frac {6 c^6 \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {6 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )}{b^{5/2}}}{3 d^2} \] Input:

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

Output:

((d*(8*a^4*d^4 + 3*b^4*c^4*x^4 + a*b^3*c^2*x^2*(6*c^2 + 7*c*d*x + 6*d^2*x^ 
2) + a^3*b*d^2*(14*c^2 + 3*c*d*x + 12*d^2*x^2) + a^2*b^2*(3*c^4 + 6*c^3*d* 
x + 21*c^2*d^2*x^2 + 4*c*d^3*x^3 + 3*d^4*x^4)))/(b^3*(b*c^2 + a*d^2)^2*(a 
+ b*x^2)^(3/2)) - (6*c^6*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d 
*x) - c*Sqrt[a + b*x^2])])/(-(b*c^2) - a*d^2)^(5/2) + (6*c*ArcTanh[(Sqrt[b 
]*x)/(Sqrt[a] - Sqrt[a + b*x^2])])/b^(5/2))/(3*d^2)
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {601, 25, 2178, 27, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/3*(a^2*(a*d + b*c*x))/(b^3*(b*c^2 + a*d^2)*(a + b*x^2)^(3/2)) + ((a^2*( 
3*a*d*(3*b*c^2 + 2*a*d^2) + b*c*(7*b*c^2 + 4*a*d^2)*x))/(b^3*(b*c^2 + a*d^ 
2)^2*Sqrt[a + b*x^2]) - (3*a*(-(Sqrt[a + b*x^2]/(b*d)) + (c*(ArcTanh[(Sqrt 
[b]*x)/Sqrt[a + b*x^2]]/(Sqrt[b]*d) + (b^2*c^5*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)^(5/2))))/d))/b^2)/(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 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. \(709\) vs. \(2(196)=392\).

Time = 0.45 (sec) , antiderivative size = 710, normalized size of antiderivative = 3.26

method	result	size
risch	\(\frac {\sqrt {b \,x^{2}+a}}{b^{3} d}-\frac {\frac {c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {d \,a^{2} \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{4 b \left (\sqrt {-a b}\, d -b c \right )}-\frac {d \,a^{2} \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{4 b \left (\sqrt {-a b}\, d +b c \right )}+\frac {b^{4} c^{6} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {d \left (6 \left (-a b \right )^{\frac {3}{2}} d -5 c a \,b^{2}+2 a b d \sqrt {-a b}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{4 b^{2} \left (\sqrt {-a b}\, d +b c \right )^{2} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \left (6 \left (-a b \right )^{\frac {3}{2}} d +5 c a \,b^{2}+2 a b d \sqrt {-a b}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{4 b^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} \left (x +\frac {\sqrt {-a b}}{b}\right )}}{b^{2} d}\)	\(710\)
default	\(\frac {\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}}{d}+\frac {c^{2} \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{d^{3}}-\frac {c^{4}}{3 d^{5} b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {c^{6} \left (\frac {d^{2}}{3 \left (a \,d^{2}+b \,c^{2}\right ) \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}+\frac {b c d \left (\frac {\frac {4 b \left (x +\frac {c}{d}\right )}{3}-\frac {4 b c}{3 d}}{\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}+\frac {16 b \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{3 {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right )}^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {d^{2} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}\right )}{d^{7}}-\frac {c^{5} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{d^{6}}-\frac {c \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{d^{2}}-\frac {c^{3} \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )}{d^{4}}\)	\(889\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 696 vs. \(2 (197) = 394\).

Time = 33.61 (sec) , antiderivative size = 2849, normalized size of antiderivative = 13.07 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6*(3*(a^2*b^3*c^7 + 3*a^3*b^2*c^5*d^2 + 3*a^4*b*c^3*d^4 + a^5*c*d^6 + ( 
b^5*c^7 + 3*a*b^4*c^5*d^2 + 3*a^2*b^3*c^3*d^4 + a^3*b^2*c*d^6)*x^4 + 2*(a* 
b^4*c^7 + 3*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^3*d^4 + a^4*b*c*d^6)*x^2)*sqrt(b 
)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(b^5*c^6*x^4 + 2*a*b 
^4*c^6*x^2 + a^2*b^3*c^6)*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*(3*a^2*b^3*c^6*d + 17*a 
^3*b^2*c^4*d^3 + 22*a^4*b*c^2*d^5 + 8*a^5*d^7 + 3*(b^5*c^6*d + 3*a*b^4*c^4 
*d^3 + 3*a^2*b^3*c^2*d^5 + a^3*b^2*d^7)*x^4 + (7*a*b^4*c^5*d^2 + 11*a^2*b^ 
3*c^3*d^4 + 4*a^3*b^2*c*d^6)*x^3 + 3*(2*a*b^4*c^6*d + 9*a^2*b^3*c^4*d^3 + 
11*a^3*b^2*c^2*d^5 + 4*a^4*b*d^7)*x^2 + 3*(2*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c 
^3*d^4 + a^4*b*c*d^6)*x)*sqrt(b*x^2 + a))/(a^2*b^6*c^6*d^2 + 3*a^3*b^5*c^4 
*d^4 + 3*a^4*b^4*c^2*d^6 + a^5*b^3*d^8 + (b^8*c^6*d^2 + 3*a*b^7*c^4*d^4 + 
3*a^2*b^6*c^2*d^6 + a^3*b^5*d^8)*x^4 + 2*(a*b^7*c^6*d^2 + 3*a^2*b^6*c^4*d^ 
4 + 3*a^3*b^5*c^2*d^6 + a^4*b^4*d^8)*x^2), -1/6*(6*(b^5*c^6*x^4 + 2*a*b^4* 
c^6*x^2 + a^2*b^3*c^6)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b 
*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) 
 - 3*(a^2*b^3*c^7 + 3*a^3*b^2*c^5*d^2 + 3*a^4*b*c^3*d^4 + a^5*c*d^6 + (b^5 
*c^7 + 3*a*b^4*c^5*d^2 + 3*a^2*b^3*c^3*d^4 + a^3*b^2*c*d^6)*x^4 + 2*(a*b^4 
*c^7 + 3*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^3*d^4 + a^4*b*c*d^6)*x^2)*sqrt(b...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (197) = 394\).

Time = 0.27 (sec) , antiderivative size = 760, normalized size of antiderivative = 3.49 \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

b*c^7*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^4 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^6 + 
 sqrt(b*x^2 + a)*a^3*d^8) + 1/3*b*c^7*x/((b*x^2 + a)^(3/2)*a*b*c^2*d^6 + ( 
b*x^2 + a)^(3/2)*a^2*d^8) + 2/3*b*c^7*x/(sqrt(b*x^2 + a)*a^2*b*c^2*d^6 + s 
qrt(b*x^2 + a)*a^3*d^8) + c^6/(sqrt(b*x^2 + a)*b^2*c^4*d^3 + 2*sqrt(b*x^2 
+ a)*a*b*c^2*d^5 + sqrt(b*x^2 + a)*a^2*d^7) + 1/3*c^6/((b*x^2 + a)^(3/2)*b 
*c^2*d^5 + (b*x^2 + a)^(3/2)*a*d^7) + x^4/((b*x^2 + a)^(3/2)*b*d) + 5*c*x^ 
3/((b*x^2 + a)^(3/2)*b*d^2) - 2/3*c^4*x^2/(sqrt(b*x^2 + a)*a^2*d^5) - 1/3* 
c^4*x^2/((b*x^2 + a)^(3/2)*a*d^5) - 3*c^2*x^2/((b*x^2 + a)^(3/2)*b*d^3) - 
c^2*x^2/(sqrt(b*x^2 + a)*a*b*d^3) + 4*a*x^2/((b*x^2 + a)^(3/2)*b^2*d) - 2/ 
3*c^5*x/(sqrt(b*x^2 + a)*a^2*d^6) - 1/3*c^5*x/((b*x^2 + a)^(3/2)*a*d^6) + 
1/3*c^3*x/((b*x^2 + a)^(3/2)*b*d^4) - 1/3*c^3*x/(sqrt(b*x^2 + a)*a*b*d^4) 
- 11/3*c*x/(sqrt(b*x^2 + a)*b^2*d^2) + 14/3*a*c*x/((b*x^2 + a)^(3/2)*b^2*d 
^2) - c*arcsinh(b*x/sqrt(a*b))/(b^(5/2)*d^2) + 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^2/d^2)^(5/2)*d^7 
) - 2/3*c^4/((b*x^2 + a)^(3/2)*b*d^5) + 2/3*sqrt(b*x^2 + a)*c^4/(a^2*b*d^5 
) - 1/3*c^4/(sqrt(b*x^2 + a)*a*b*d^5) + c^2/(sqrt(b*x^2 + a)*b^2*d^3) + sq 
rt(b*x^2 + a)*c^2/(a*b^2*d^3) - 8/3*a*c^2/((b*x^2 + a)^(3/2)*b^2*d^3) + 8/ 
3*a^2/((b*x^2 + a)^(3/2)*b^3*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^6}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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