\(\int \frac {x^6 (c+d x)}{(a+b x^2)^{7/2}} \, dx\) [1307]

Optimal result

Integrand size = 20, antiderivative size = 130 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {a (15 a d-11 b c x)}{15 b^4 \left (a+b x^2\right )^{3/2}}+\frac {45 a d-23 b c x}{15 b^4 \sqrt {a+b x^2}}+\frac {d \sqrt {a+b x^2}}{b^4}+\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{7/2}} \] Output:

1/5*a^2*(-b*c*x+a*d)/b^4/(b*x^2+a)^(5/2)-1/15*a*(-11*b*c*x+15*a*d)/b^4/(b* 
x^2+a)^(3/2)+1/15*(-23*b*c*x+45*a*d)/b^4/(b*x^2+a)^(1/2)+d*(b*x^2+a)^(1/2) 
/b^4+c*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {48 a^3 d-15 a^2 b c x+120 a^2 b d x^2-35 a b^2 c x^3+90 a b^2 d x^4-23 b^3 c x^5+15 b^3 d x^6}{15 b^4 \left (a+b x^2\right )^{5/2}}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{7/2}} \] Input:

Integrate[(x^6*(c + d*x))/(a + b*x^2)^(7/2),x]
 

Output:

(48*a^3*d - 15*a^2*b*c*x + 120*a^2*b*d*x^2 - 35*a*b^2*c*x^3 + 90*a*b^2*d*x 
^4 - 23*b^3*c*x^5 + 15*b^3*d*x^6)/(15*b^4*(a + b*x^2)^(5/2)) + (2*c*ArcTan 
h[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/b^(7/2)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {530, 25, 2345, 2345, 27, 455, 224, 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.

\(\displaystyle \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 530

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {-\frac {\int \frac {\frac {8 c a^3}{b^3}+\frac {30 d x a^3}{b^3}-\frac {15 d x^3 a^2}{b^2}-\frac {15 c x^2 a^2}{b^2}}{\left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {-\frac {-\frac {\int \frac {15 a^3 (c+d x)}{b^3 \sqrt {b x^2+a}}dx}{a}-\frac {a^2 (45 a d-23 b c x)}{b^4 \sqrt {a+b x^2}}}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {15 a^2 \int \frac {c+d x}{\sqrt {b x^2+a}}dx}{b^3}-\frac {a^2 (45 a d-23 b c x)}{b^4 \sqrt {a+b x^2}}}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {-\frac {-\frac {15 a^2 \left (c \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {d \sqrt {a+b x^2}}{b}\right )}{b^3}-\frac {a^2 (45 a d-23 b c x)}{b^4 \sqrt {a+b x^2}}}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {-\frac {15 a^2 \left (c \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {d \sqrt {a+b x^2}}{b}\right )}{b^3}-\frac {a^2 (45 a d-23 b c x)}{b^4 \sqrt {a+b x^2}}}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {15 a^2 \left (\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {d \sqrt {a+b x^2}}{b}\right )}{b^3}-\frac {a^2 (45 a d-23 b c x)}{b^4 \sqrt {a+b x^2}}}{3 a}-\frac {a^2 (15 a d-11 b c x)}{3 b^4 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {a^2 (a d-b c x)}{5 b^4 \left (a+b x^2\right )^{5/2}}\)

Input:

Int[(x^6*(c + d*x))/(a + b*x^2)^(7/2),x]
 

Output:

(a^2*(a*d - b*c*x))/(5*b^4*(a + b*x^2)^(5/2)) + (-1/3*(a^2*(15*a*d - 11*b* 
c*x))/(b^4*(a + b*x^2)^(3/2)) - (-((a^2*(45*a*d - 23*b*c*x))/(b^4*Sqrt[a + 
 b*x^2])) - (15*a^2*((d*Sqrt[a + b*x^2])/b + (c*ArcTanh[(Sqrt[b]*x)/Sqrt[a 
 + b*x^2]])/Sqrt[b]))/b^3)/(3*a))/(5*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 455

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

rule 530

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb 
ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co 
eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po 
lynomialRemainder[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[(a 
 + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; 
FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 
1] && IntegerQ[2*p]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.28

method	result	size
default	\(c \left (-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}\right )+d \left (\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {6 a \left (-\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {4 a \left (-\frac {x^{2}}{3 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {2 a}{15 b^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}\right )}{b}\right )}{b}\right )\)	\(166\)
risch	\(\frac {d \sqrt {b \,x^{2}+a}}{b^{4}}+\frac {c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {7}{2}}}+\frac {3 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, \sqrt {-a b}\, d}{2 b^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {23 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c}{30 b^{4} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {3 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, \sqrt {-a b}\, d}{2 b^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {23 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c}{30 b^{4} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {41 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c}{240 b^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {17 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, d}{80 b^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {17 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, d}{80 b^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {41 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c}{240 b^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {17 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, d}{80 b^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {17 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, d}{80 b^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, d}{40 b^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c}{40 b^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, d}{40 b^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c}{40 b^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}\)	\(946\)

Input:

int(x^6*(d*x+c)/(b*x^2+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.71 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\left [\frac {15 \, {\left (b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (15 \, b^{3} d x^{6} - 23 \, b^{3} c x^{5} + 90 \, a b^{2} d x^{4} - 35 \, a b^{2} c x^{3} + 120 \, a^{2} b d x^{2} - 15 \, a^{2} b c x + 48 \, a^{3} d\right )} \sqrt {b x^{2} + a}}{30 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, -\frac {15 \, {\left (b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (15 \, b^{3} d x^{6} - 23 \, b^{3} c x^{5} + 90 \, a b^{2} d x^{4} - 35 \, a b^{2} c x^{3} + 120 \, a^{2} b d x^{2} - 15 \, a^{2} b c x + 48 \, a^{3} d\right )} \sqrt {b x^{2} + a}}{15 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="fricas")
 

Output:

[1/30*(15*(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(b)*log( 
-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(15*b^3*d*x^6 - 23*b^3*c*x 
^5 + 90*a*b^2*d*x^4 - 35*a*b^2*c*x^3 + 120*a^2*b*d*x^2 - 15*a^2*b*c*x + 48 
*a^3*d)*sqrt(b*x^2 + a))/(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4) 
, -1/15*(15*(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(-b)*a 
rctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (15*b^3*d*x^6 - 23*b^3*c*x^5 + 90*a*b^ 
2*d*x^4 - 35*a*b^2*c*x^3 + 120*a^2*b*d*x^2 - 15*a^2*b*c*x + 48*a^3*d)*sqrt 
(b*x^2 + a))/(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1049 vs. \(2 (122) = 244\).

Time = 12.07 (sec) , antiderivative size = 1326, normalized size of antiderivative = 10.20 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(x**6*(d*x+c)/(b*x**2+a)**(7/2),x)
 

Output:

c*(15*a**(99/2)*b**25*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(15*a**( 
99/2)*b**(57/2)*sqrt(1 + b*x**2/a) + 45*a**(97/2)*b**(59/2)*x**2*sqrt(1 + 
b*x**2/a) + 45*a**(95/2)*b**(61/2)*x**4*sqrt(1 + b*x**2/a) + 15*a**(93/2)* 
b**(63/2)*x**6*sqrt(1 + b*x**2/a)) + 45*a**(97/2)*b**26*x**2*sqrt(1 + b*x* 
*2/a)*asinh(sqrt(b)*x/sqrt(a))/(15*a**(99/2)*b**(57/2)*sqrt(1 + b*x**2/a) 
+ 45*a**(97/2)*b**(59/2)*x**2*sqrt(1 + b*x**2/a) + 45*a**(95/2)*b**(61/2)* 
x**4*sqrt(1 + b*x**2/a) + 15*a**(93/2)*b**(63/2)*x**6*sqrt(1 + b*x**2/a)) 
+ 45*a**(95/2)*b**27*x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(15* 
a**(99/2)*b**(57/2)*sqrt(1 + b*x**2/a) + 45*a**(97/2)*b**(59/2)*x**2*sqrt( 
1 + b*x**2/a) + 45*a**(95/2)*b**(61/2)*x**4*sqrt(1 + b*x**2/a) + 15*a**(93 
/2)*b**(63/2)*x**6*sqrt(1 + b*x**2/a)) + 15*a**(93/2)*b**28*x**6*sqrt(1 + 
b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(15*a**(99/2)*b**(57/2)*sqrt(1 + b*x**2 
/a) + 45*a**(97/2)*b**(59/2)*x**2*sqrt(1 + b*x**2/a) + 45*a**(95/2)*b**(61 
/2)*x**4*sqrt(1 + b*x**2/a) + 15*a**(93/2)*b**(63/2)*x**6*sqrt(1 + b*x**2/ 
a)) - 15*a**49*b**(51/2)*x/(15*a**(99/2)*b**(57/2)*sqrt(1 + b*x**2/a) + 45 
*a**(97/2)*b**(59/2)*x**2*sqrt(1 + b*x**2/a) + 45*a**(95/2)*b**(61/2)*x**4 
*sqrt(1 + b*x**2/a) + 15*a**(93/2)*b**(63/2)*x**6*sqrt(1 + b*x**2/a)) - 50 
*a**48*b**(53/2)*x**3/(15*a**(99/2)*b**(57/2)*sqrt(1 + b*x**2/a) + 45*a**( 
97/2)*b**(59/2)*x**2*sqrt(1 + b*x**2/a) + 45*a**(95/2)*b**(61/2)*x**4*sqrt 
(1 + b*x**2/a) + 15*a**(93/2)*b**(63/2)*x**6*sqrt(1 + b*x**2/a)) - 58*a...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.69 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {d x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} - \frac {1}{15} \, c x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )} - \frac {c x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b} + \frac {6 \, a d x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2} d x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} + \frac {7 \, c x}{15 \, \sqrt {b x^{2} + a} b^{3}} - \frac {4 \, a c x}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {7}{2}}} + \frac {16 \, a^{3} d}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="maxima")
 

Output:

d*x^6/((b*x^2 + a)^(5/2)*b) - 1/15*c*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 20* 
a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3)) - 1/3*c*x*( 
3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b + 6*a*d*x^4/( 
(b*x^2 + a)^(5/2)*b^2) + 8*a^2*d*x^2/((b*x^2 + a)^(5/2)*b^3) + 7/15*c*x/(s 
qrt(b*x^2 + a)*b^3) - 4/15*a*c*x/((b*x^2 + a)^(3/2)*b^3) + c*arcsinh(b*x/s 
qrt(a*b))/b^(7/2) + 16/5*a^3*d/((b*x^2 + a)^(5/2)*b^4)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {{\left ({\left ({\left ({\left ({\left (\frac {15 \, d x}{b} - \frac {23 \, c}{b}\right )} x + \frac {90 \, a d}{b^{2}}\right )} x - \frac {35 \, a c}{b^{2}}\right )} x + \frac {120 \, a^{2} d}{b^{3}}\right )} x - \frac {15 \, a^{2} c}{b^{3}}\right )} x + \frac {48 \, a^{3} d}{b^{4}}}{15 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}}} - \frac {c \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {7}{2}}} \] Input:

integrate(x^6*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="giac")
 

Output:

1/15*((((((15*d*x/b - 23*c/b)*x + 90*a*d/b^2)*x - 35*a*c/b^2)*x + 120*a^2* 
d/b^3)*x - 15*a^2*c/b^3)*x + 48*a^3*d/b^4)/(b*x^2 + a)^(5/2) - c*log(abs(- 
sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\int \frac {x^6\,\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^{7/2}} \,d x \] Input:

int((x^6*(c + d*x))/(a + b*x^2)^(7/2),x)
 

Output:

int((x^6*(c + d*x))/(a + b*x^2)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.44 \[ \int \frac {x^6 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {48 \sqrt {b \,x^{2}+a}\, a^{3} d -15 \sqrt {b \,x^{2}+a}\, a^{2} b c x +120 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{2}-35 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{3}+90 \sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{4}-23 \sqrt {b \,x^{2}+a}\, b^{3} c \,x^{5}+15 \sqrt {b \,x^{2}+a}\, b^{3} d \,x^{6}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} c +45 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c \,x^{2}+45 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c \,x^{4}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{6}+5 \sqrt {b}\, a^{3} c +15 \sqrt {b}\, a^{2} b c \,x^{2}+15 \sqrt {b}\, a \,b^{2} c \,x^{4}+5 \sqrt {b}\, b^{3} c \,x^{6}}{15 b^{4} \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:

int(x^6*(d*x+c)/(b*x^2+a)^(7/2),x)
 

Output:

(48*sqrt(a + b*x**2)*a**3*d - 15*sqrt(a + b*x**2)*a**2*b*c*x + 120*sqrt(a 
+ b*x**2)*a**2*b*d*x**2 - 35*sqrt(a + b*x**2)*a*b**2*c*x**3 + 90*sqrt(a + 
b*x**2)*a*b**2*d*x**4 - 23*sqrt(a + b*x**2)*b**3*c*x**5 + 15*sqrt(a + b*x* 
*2)*b**3*d*x**6 + 15*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a 
**3*c + 45*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*c*x* 
*2 + 45*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*c*x**4 
+ 15*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**3*c*x**6 + 5*s 
qrt(b)*a**3*c + 15*sqrt(b)*a**2*b*c*x**2 + 15*sqrt(b)*a*b**2*c*x**4 + 5*sq 
rt(b)*b**3*c*x**6)/(15*b**4*(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x 
**6))