Integrand size = 20, antiderivative size = 149 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {a^2 (15 c+16 d x)}{15 b^4 \left (a+b x^2\right )^{3/2}}+\frac {a (45 c+58 d x)}{15 b^4 \sqrt {a+b x^2}}+\frac {c \sqrt {a+b x^2}}{b^4}+\frac {d x \sqrt {a+b x^2}}{2 b^4}-\frac {7 a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{9/2}} \] Output:
1/5*a^3*(d*x+c)/b^4/(b*x^2+a)^(5/2)-1/15*a^2*(16*d*x+15*c)/b^4/(b*x^2+a)^( 3/2)+1/15*a*(58*d*x+45*c)/b^4/(b*x^2+a)^(1/2)+c*(b*x^2+a)^(1/2)/b^4+1/2*d* x*(b*x^2+a)^(1/2)/b^4-7/2*a*d*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
Time = 0.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {15 b^3 x^6 (2 c+d x)+3 a^3 (32 c+35 d x)+5 a^2 b x^2 (48 c+49 d x)+a b^2 x^4 (180 c+161 d x)}{30 b^4 \left (a+b x^2\right )^{5/2}}+\frac {7 a d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{9/2}} \] Input:
Integrate[(x^7*(c + d*x))/(a + b*x^2)^(7/2),x]
Output:
(15*b^3*x^6*(2*c + d*x) + 3*a^3*(32*c + 35*d*x) + 5*a^2*b*x^2*(48*c + 49*d *x) + a*b^2*x^4*(180*c + 161*d*x))/(30*b^4*(a + b*x^2)^(5/2)) + (7*a*d*Log [-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^(9/2))
Time = 1.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {530, 2345, 2345, 27, 2346, 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.
| \(\displaystyle \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {-\frac {5 a d x^6}{b}-\frac {5 a c x^5}{b}+\frac {5 a^2 d x^4}{b^2}+\frac {5 a^2 c x^3}{b^2}-\frac {5 a^3 d x^2}{b^3}-\frac {5 a^3 c x}{b^3}+\frac {a^4 d}{b^4}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\frac {13 d a^4}{b^4}-\frac {30 d x^2 a^3}{b^3}-\frac {30 c x a^3}{b^3}+\frac {15 d x^4 a^2}{b^2}+\frac {15 c x^3 a^2}{b^2}}{\left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {\int \frac {15 \left (\frac {3 d a^4}{b^4}-\frac {d x^2 a^3}{b^3}-\frac {c x a^3}{b^3}\right )}{\sqrt {b x^2+a}}dx}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \int \frac {\frac {3 d a^4}{b^4}-\frac {d x^2 a^3}{b^3}-\frac {c x a^3}{b^3}}{\sqrt {b x^2+a}}dx}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \left (\frac {\int \frac {a^3 (7 a d-2 b c x)}{b^3 \sqrt {b x^2+a}}dx}{2 b}-\frac {a^3 d x \sqrt {a+b x^2}}{2 b^4}\right )}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \left (\frac {a^3 \int \frac {7 a d-2 b c x}{\sqrt {b x^2+a}}dx}{2 b^4}-\frac {a^3 d x \sqrt {a+b x^2}}{2 b^4}\right )}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \left (\frac {a^3 \left (7 a d \int \frac {1}{\sqrt {b x^2+a}}dx-2 c \sqrt {a+b x^2}\right )}{2 b^4}-\frac {a^3 d x \sqrt {a+b x^2}}{2 b^4}\right )}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \left (\frac {a^3 \left (7 a d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-2 c \sqrt {a+b x^2}\right )}{2 b^4}-\frac {a^3 d x \sqrt {a+b x^2}}{2 b^4}\right )}{a}}{3 a}}{5 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a^3 (c+d x)}{5 b^4 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a^3 (15 c+16 d x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 c+58 d x)}{b^4 \sqrt {a+b x^2}}-\frac {15 \left (\frac {a^3 \left (\frac {7 a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-2 c \sqrt {a+b x^2}\right )}{2 b^4}-\frac {a^3 d x \sqrt {a+b x^2}}{2 b^4}\right )}{a}}{3 a}}{5 a}\) |
Input:
Int[(x^7*(c + d*x))/(a + b*x^2)^(7/2),x]
Output:
(a^3*(c + d*x))/(5*b^4*(a + b*x^2)^(5/2)) - ((a^3*(15*c + 16*d*x))/(3*b^4* (a + b*x^2)^(3/2)) - ((a^3*(45*c + 58*d*x))/(b^4*Sqrt[a + b*x^2]) - (15*(- 1/2*(a^3*d*x*Sqrt[a + b*x^2])/b^4 + (a^3*(-2*c*Sqrt[a + b*x^2] + (7*a*d*Ar cTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]))/(2*b^4)))/a)/(3*a))/(5*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.48 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(c \left (\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {6 a \left (-\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {4 a \left (-\frac {x^{2}}{3 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {2 a}{15 b^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}\right )}{b}\right )}{b}\right )+d \left (\frac {x^{7}}{2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {7 a \left (-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}\right )}{2 b}\right )\) | \(190\) |
| risch | \(\frac {\left (d x +2 c \right ) \sqrt {b \,x^{2}+a}}{2 b^{4}}-\frac {7 a d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {9}{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}\, c}{2 b^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {29 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}{15 b^{5} \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}\, c}{2 b^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {29 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}{15 b^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {61 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}{240 b^{5} \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 )}\, c}{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 )}\, c}{80 b^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {61 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}{240 b^{5} \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 )}\, c}{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 )}\, c}{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 )}\, c}{40 b^{5} \sqrt {-a b}\, \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^{6} \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 )}\, c}{40 b^{5} \sqrt {-a b}\, \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^{6} \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}\) | \(965\) |
Input:
int(x^7*(d*x+c)/(b*x^2+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
c*(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))))+d*(1/2*x^7/b/(b*x^2+a)^(5/2)- 7/2*a/b*(-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))))))
Time = 0.12 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.60 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\left [\frac {105 \, {\left (a b^{3} d x^{6} + 3 \, a^{2} b^{2} d x^{4} + 3 \, a^{3} b d x^{2} + a^{4} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (15 \, b^{4} d x^{7} + 30 \, b^{4} c x^{6} + 161 \, a b^{3} d x^{5} + 180 \, a b^{3} c x^{4} + 245 \, a^{2} b^{2} d x^{3} + 240 \, a^{2} b^{2} c x^{2} + 105 \, a^{3} b d x + 96 \, a^{3} b c\right )} \sqrt {b x^{2} + a}}{60 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac {105 \, {\left (a b^{3} d x^{6} + 3 \, a^{2} b^{2} d x^{4} + 3 \, a^{3} b d x^{2} + a^{4} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, b^{4} d x^{7} + 30 \, b^{4} c x^{6} + 161 \, a b^{3} d x^{5} + 180 \, a b^{3} c x^{4} + 245 \, a^{2} b^{2} d x^{3} + 240 \, a^{2} b^{2} c x^{2} + 105 \, a^{3} b d x + 96 \, a^{3} b c\right )} \sqrt {b x^{2} + a}}{30 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \] Input:
integrate(x^7*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="fricas")
Output:
[1/60*(105*(a*b^3*d*x^6 + 3*a^2*b^2*d*x^4 + 3*a^3*b*d*x^2 + a^4*d)*sqrt(b) *log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(15*b^4*d*x^7 + 30*b^ 4*c*x^6 + 161*a*b^3*d*x^5 + 180*a*b^3*c*x^4 + 245*a^2*b^2*d*x^3 + 240*a^2* b^2*c*x^2 + 105*a^3*b*d*x + 96*a^3*b*c)*sqrt(b*x^2 + a))/(b^8*x^6 + 3*a*b^ 7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5), 1/30*(105*(a*b^3*d*x^6 + 3*a^2*b^2*d*x^4 + 3*a^3*b*d*x^2 + a^4*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (1 5*b^4*d*x^7 + 30*b^4*c*x^6 + 161*a*b^3*d*x^5 + 180*a*b^3*c*x^4 + 245*a^2*b ^2*d*x^3 + 240*a^2*b^2*c*x^2 + 105*a^3*b*d*x + 96*a^3*b*c)*sqrt(b*x^2 + a) )/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)]
Time = 16.17 (sec) , antiderivative size = 1445, normalized size of antiderivative = 9.70 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(x**7*(d*x+c)/(b*x**2+a)**(7/2),x)
Output:
c*Piecewise((16*a**3/(5*a**2*b**4*sqrt(a + b*x**2) + 10*a*b**5*x**2*sqrt(a + b*x**2) + 5*b**6*x**4*sqrt(a + b*x**2)) + 40*a**2*b*x**2/(5*a**2*b**4*s qrt(a + b*x**2) + 10*a*b**5*x**2*sqrt(a + b*x**2) + 5*b**6*x**4*sqrt(a + b *x**2)) + 30*a*b**2*x**4/(5*a**2*b**4*sqrt(a + b*x**2) + 10*a*b**5*x**2*sq rt(a + b*x**2) + 5*b**6*x**4*sqrt(a + b*x**2)) + 5*b**3*x**6/(5*a**2*b**4* sqrt(a + b*x**2) + 10*a*b**5*x**2*sqrt(a + b*x**2) + 5*b**6*x**4*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(7/2)), True)) + d*(-105*a**(169/2)*b**41 *sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(30*a**(167/2)*b**(91/2)*sqrt (1 + b*x**2/a) + 90*a**(165/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a) + 90*a**( 163/2)*b**(95/2)*x**4*sqrt(1 + b*x**2/a) + 30*a**(161/2)*b**(97/2)*x**6*sq rt(1 + b*x**2/a)) - 315*a**(167/2)*b**42*x**2*sqrt(1 + b*x**2/a)*asinh(sqr t(b)*x/sqrt(a))/(30*a**(167/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 90*a**(165/2 )*b**(93/2)*x**2*sqrt(1 + b*x**2/a) + 90*a**(163/2)*b**(95/2)*x**4*sqrt(1 + b*x**2/a) + 30*a**(161/2)*b**(97/2)*x**6*sqrt(1 + b*x**2/a)) - 315*a**(1 65/2)*b**43*x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(30*a**(167/2 )*b**(91/2)*sqrt(1 + b*x**2/a) + 90*a**(165/2)*b**(93/2)*x**2*sqrt(1 + b*x **2/a) + 90*a**(163/2)*b**(95/2)*x**4*sqrt(1 + b*x**2/a) + 30*a**(161/2)*b **(97/2)*x**6*sqrt(1 + b*x**2/a)) - 105*a**(163/2)*b**44*x**6*sqrt(1 + b*x **2/a)*asinh(sqrt(b)*x/sqrt(a))/(30*a**(167/2)*b**(91/2)*sqrt(1 + b*x**2/a ) + 90*a**(165/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a) + 90*a**(163/2)*b**...
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (123) = 246\).
Time = 0.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.66 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {d x^{7}}{2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {c x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {7 \, a d x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{30 \, b} + \frac {7 \, a d x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b^{2}} + \frac {6 \, a c x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2} c x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {49 \, a d x}{30 \, \sqrt {b x^{2} + a} b^{4}} + \frac {14 \, a^{2} d x}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {7 \, a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {9}{2}}} + \frac {16 \, a^{3} c}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} \] Input:
integrate(x^7*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="maxima")
Output:
1/2*d*x^7/((b*x^2 + a)^(5/2)*b) + c*x^6/((b*x^2 + a)^(5/2)*b) + 7/30*a*d*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))/b + 7/6*a*d*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/ ((b*x^2 + a)^(3/2)*b^2))/b^2 + 6*a*c*x^4/((b*x^2 + a)^(5/2)*b^2) + 8*a^2*c *x^2/((b*x^2 + a)^(5/2)*b^3) - 49/30*a*d*x/(sqrt(b*x^2 + a)*b^4) + 14/15*a ^2*d*x/((b*x^2 + a)^(3/2)*b^4) - 7/2*a*d*arcsinh(b*x/sqrt(a*b))/b^(9/2) + 16/5*a^3*c/((b*x^2 + a)^(5/2)*b^4)
Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.80 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {{\left ({\left ({\left ({\left ({\left (15 \, {\left (\frac {d x}{b} + \frac {2 \, c}{b}\right )} x + \frac {161 \, a d}{b^{2}}\right )} x + \frac {180 \, a c}{b^{2}}\right )} x + \frac {245 \, a^{2} d}{b^{3}}\right )} x + \frac {240 \, a^{2} c}{b^{3}}\right )} x + \frac {105 \, a^{3} d}{b^{4}}\right )} x + \frac {96 \, a^{3} c}{b^{4}}}{30 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}}} + \frac {7 \, a d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {9}{2}}} \] Input:
integrate(x^7*(d*x+c)/(b*x^2+a)^(7/2),x, algorithm="giac")
Output:
1/30*((((((15*(d*x/b + 2*c/b)*x + 161*a*d/b^2)*x + 180*a*c/b^2)*x + 245*a^ 2*d/b^3)*x + 240*a^2*c/b^3)*x + 105*a^3*d/b^4)*x + 96*a^3*c/b^4)/(b*x^2 + a)^(5/2) + 7/2*a*d*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\int \frac {x^7\,\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^{7/2}} \,d x \] Input:
int((x^7*(c + d*x))/(a + b*x^2)^(7/2),x)
Output:
int((x^7*(c + d*x))/(a + b*x^2)^(7/2), x)
Time = 0.23 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.32 \[ \int \frac {x^7 (c+d x)}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {384 \sqrt {b \,x^{2}+a}\, a^{3} b c +420 \sqrt {b \,x^{2}+a}\, a^{3} b d x +960 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{2}+980 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{3}+720 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{4}+644 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{5}+120 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{6}+60 \sqrt {b \,x^{2}+a}\, b^{4} d \,x^{7}-420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d -1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d \,x^{2}-1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,x^{4}-420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,x^{6}-203 \sqrt {b}\, a^{4} d -609 \sqrt {b}\, a^{3} b d \,x^{2}-609 \sqrt {b}\, a^{2} b^{2} d \,x^{4}-203 \sqrt {b}\, a \,b^{3} d \,x^{6}}{120 b^{5} \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:
int(x^7*(d*x+c)/(b*x^2+a)^(7/2),x)
Output:
(384*sqrt(a + b*x**2)*a**3*b*c + 420*sqrt(a + b*x**2)*a**3*b*d*x + 960*sqr t(a + b*x**2)*a**2*b**2*c*x**2 + 980*sqrt(a + b*x**2)*a**2*b**2*d*x**3 + 7 20*sqrt(a + b*x**2)*a*b**3*c*x**4 + 644*sqrt(a + b*x**2)*a*b**3*d*x**5 + 1 20*sqrt(a + b*x**2)*b**4*c*x**6 + 60*sqrt(a + b*x**2)*b**4*d*x**7 - 420*sq rt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d - 1260*sqrt(b)*lo g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d*x**2 - 1260*sqrt(b)*log ((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d*x**4 - 420*sqrt(b)*lo g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**3*d*x**6 - 203*sqrt(b)*a**4 *d - 609*sqrt(b)*a**3*b*d*x**2 - 609*sqrt(b)*a**2*b**2*d*x**4 - 203*sqrt(b )*a*b**3*d*x**6)/(120*b**5*(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x* *6))