Integrand size = 24, antiderivative size = 332 \[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}-\frac {d^3 \sqrt {d x} (5 A+7 B x)}{6 b^2 \sqrt {a+b x^2}}+\frac {7 B d^3 \sqrt {d x} \sqrt {a+b x^2}}{2 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {7 \sqrt [4]{a} B d^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{2 b^{11/4} \sqrt {a+b x^2}}+\frac {\left (5 A \sqrt {b}+21 \sqrt {a} B\right ) d^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{11/4} \sqrt {a+b x^2}} \] Output:
-1/3*d*(d*x)^(5/2)*(B*x+A)/b/(b*x^2+a)^(3/2)-1/6*d^3*(d*x)^(1/2)*(7*B*x+5* A)/b^2/(b*x^2+a)^(1/2)+7/2*B*d^3*(d*x)^(1/2)*(b*x^2+a)^(1/2)/b^(5/2)/(a^(1 /2)+b^(1/2)*x)-7/2*a^(1/4)*B*d^(7/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/ 2)+b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/ d^(1/2))),1/2*2^(1/2))/b^(11/4)/(b*x^2+a)^(1/2)+1/12*(5*A*b^(1/2)+21*a^(1/ 2)*B)*d^(7/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)* InverseJacobiAM(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^(1/2)) /a^(1/4)/b^(11/4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.42 \[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^3 \sqrt {d x} \left (-5 a A-7 a B x-7 A b x^2-9 b B x^3+5 A \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )+7 B x \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{6 b^2 \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[((d*x)^(7/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
(d^3*Sqrt[d*x]*(-5*a*A - 7*a*B*x - 7*A*b*x^2 - 9*b*B*x^3 + 5*A*(a + b*x^2) *Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)] + 7*B* x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^ 2)/a)]))/(6*b^2*(a + b*x^2)^(3/2))
Time = 0.47 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {549, 27, 549, 27, 556, 555, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {d^2 \int \frac {(d x)^{3/2} (5 A+7 B x)}{2 \left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {(d x)^{3/2} (5 A+7 B x)}{\left (b x^2+a\right )^{3/2}}dx}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {5 A+21 B x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{b}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {5 A+21 B x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{2 b}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \int \frac {5 A+21 B x}{\sqrt {x} \sqrt {b x^2+a}}dx}{2 b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \int \frac {5 A+21 B x}{\sqrt {b x^2+a}}d\sqrt {x}}{b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \left (\left (\frac {21 \sqrt {a} B}{\sqrt {b}}+5 A\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {21 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \left (\left (\frac {21 \sqrt {a} B}{\sqrt {b}}+5 A\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {21 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (\frac {21 \sqrt {a} B}{\sqrt {b}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {21 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (\frac {21 \sqrt {a} B}{\sqrt {b}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {21 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {b}}\right )}{b \sqrt {d x}}-\frac {d \sqrt {d x} (5 A+7 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{5/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((d*x)^(7/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
-1/3*(d*(d*x)^(5/2)*(A + B*x))/(b*(a + b*x^2)^(3/2)) + (d^2*(-((d*Sqrt[d*x ]*(5*A + 7*B*x))/(b*Sqrt[a + b*x^2])) + (d^2*Sqrt[x]*((-21*B*(-((Sqrt[x]*S qrt[a + b*x^2])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sq rt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x ])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])))/Sqrt[b] + ((5*A + (21*Sqrt[ a]*B)/Sqrt[b])*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x )^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/ 4)*Sqrt[a + b*x^2])))/(b*Sqrt[d*x])))/(6*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[e*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b *x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free Q[{a, c, f, g}, x]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb ol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] , x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 1.46 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.20
| method | result | size |
| elliptic | \(\frac {\sqrt {d x}\, \sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (\frac {\left (\frac {a \,d^{3} B x}{3 b^{4}}+\frac {a \,d^{3} A}{3 b^{4}}\right ) \sqrt {b d \,x^{3}+a d x}}{\left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {2 b d x \left (\frac {3 d^{3} B x}{4 b^{3}}+\frac {7 d^{3} A}{12 b^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b d x}}+\frac {5 A \,d^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 b^{3} \sqrt {b d \,x^{3}+a d x}}+\frac {7 B \,d^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 b^{3} \sqrt {b d \,x^{3}+a d x}}\right )}{d x \sqrt {b \,x^{2}+a}}\) | \(400\) |
| default | \(\frac {\left (5 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b \,x^{2}+42 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-21 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+5 A \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a +42 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-21 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-18 x^{4} B \,b^{2}-14 A \,x^{3} b^{2}-14 B a \,x^{2} b -10 a b A x \right ) d^{3} \sqrt {d x}}{12 x \,b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(584\) |
Input:
int((d*x)^(7/2)*(B*x+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/d/x*(d*x)^(1/2)/(b*x^2+a)^(1/2)*(d*(b*x^2+a)*x)^(1/2)*((1/3*a*d^3/b^4*B* x+1/3*a*d^3/b^4*A)*(b*d*x^3+a*d*x)^(1/2)/(x^2+a/b)^2-2*b*d*x*(3/4*d^3*B/b^ 3*x+7/12*d^3*A/b^3)/((x^2+a/b)*b*d*x)^(1/2)+5/12*A*d^4/b^3*(-a*b)^(1/2)*(( x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2 )*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*d*x^3+a*d*x)^(1/2)*EllipticF(((x +(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+7/4*B*d^4/b^3*(-a*b)^( 1/2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a* b)^(1/2)*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*d*x^3+a*d*x)^(1/2)*(-2*(- a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/ 2))+(-a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2 *2^(1/2))))
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.57 \[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (A b^{2} d^{3} x^{4} + 2 \, A a b d^{3} x^{2} + A a^{2} d^{3}\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - 21 \, {\left (B b^{2} d^{3} x^{4} + 2 \, B a b d^{3} x^{2} + B a^{2} d^{3}\right )} \sqrt {b d} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (9 \, B b^{2} d^{3} x^{3} + 7 \, A b^{2} d^{3} x^{2} + 7 \, B a b d^{3} x + 5 \, A a b d^{3}\right )} \sqrt {b x^{2} + a} \sqrt {d x}}{6 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \] Input:
integrate((d*x)^(7/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/6*(5*(A*b^2*d^3*x^4 + 2*A*a*b*d^3*x^2 + A*a^2*d^3)*sqrt(b*d)*weierstrass PInverse(-4*a/b, 0, x) - 21*(B*b^2*d^3*x^4 + 2*B*a*b*d^3*x^2 + B*a^2*d^3)* sqrt(b*d)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) - (9*B*b^2*d^3*x^3 + 7*A*b^2*d^3*x^2 + 7*B*a*b*d^3*x + 5*A*a*b*d^3)*sqrt(b*x ^2 + a)*sqrt(d*x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)
Timed out. \[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(7/2)*(B*x+A)/(b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(7/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(d*x)^(7/2)/(b*x^2 + a)^(5/2), x)
\[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(7/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(d*x)^(7/2)/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{7/2}\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((d*x)^(7/2)*(A + B*x))/(a + b*x^2)^(5/2),x)
Output:
int(((d*x)^(7/2)*(A + B*x))/(a + b*x^2)^(5/2), x)
\[ \int \frac {(d x)^{7/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {d}\, d^{3} \left (-6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2}-6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a b \,x^{2}+14 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a b x +6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{2} x^{3}-21 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{4} b -42 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{3} b^{2} x^{2}-21 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b^{3} x^{4}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{5}+6 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{4} b \,x^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{3} b^{2} x^{4}\right )}{3 b^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((d*x)^(7/2)*(B*x+A)/(b*x^2+a)^(5/2),x)
Output:
(sqrt(d)*d**3*( - 6*sqrt(x)*sqrt(a + b*x**2)*a**2 - 6*sqrt(x)*sqrt(a + b*x **2)*a*b*x**2 + 14*sqrt(x)*sqrt(a + b*x**2)*a*b*x + 6*sqrt(x)*sqrt(a + b*x **2)*b**2*x**3 - 21*int((sqrt(x)*sqrt(a + b*x**2))/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6),x)*a**4*b - 42*int((sqrt(x)*sqrt(a + b*x**2))/ (a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6),x)*a**3*b**2*x**2 - 21* int((sqrt(x)*sqrt(a + b*x**2))/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b** 3*x**6),x)*a**2*b**3*x**4 + 3*int((sqrt(x)*sqrt(a + b*x**2))/(a**3*x + 3*a **2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7),x)*a**5 + 6*int((sqrt(x)*sqrt(a + b*x**2))/(a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7),x)*a**4*b*x* *2 + 3*int((sqrt(x)*sqrt(a + b*x**2))/(a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x **5 + b**3*x**7),x)*a**3*b**2*x**4))/(3*b**2*(a**2 + 2*a*b*x**2 + b**2*x** 4))