3.7.87 \(\int \frac {(d+e x)^9}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=206 \[ \frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 \sqrt {d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac {77 d \sqrt {d^2-e^2 x^2} (d+e x)}{2 e}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

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Rubi [A]  time = 0.10, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {669, 671, 641, 217, 203} \begin {gather*} \frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 \sqrt {d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac {77 d \sqrt {d^2-e^2 x^2} (d+e x)}{2 e}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (66*(d + e*x)^4)
/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e)
+ (77*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(5*e) - (231*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

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 641

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

Rule 669

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rubi steps

\begin {align*} \int \frac {(d+e x)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {11}{5} \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {33}{5} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}-\frac {231}{5} \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-(77 d) \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^2\right ) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 144, normalized size = 0.70 \begin {gather*} \frac {(d+e x) \left (\sqrt {1-\frac {e^2 x^2}{d^2}} \left (5446 d^5-12843 d^4 e x+8711 d^3 e^2 x^2-815 d^2 e^3 x^3-105 d e^4 x^4-10 e^5 x^5\right )-3465 d^2 (d-e x)^3 \sin ^{-1}\left (\frac {e x}{d}\right )\right )}{30 e (d-e x)^2 \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(5446*d^5 - 12843*d^4*e*x + 8711*d^3*e^2*x^2 - 815*d^2*e^3*x^3 - 105*d*e^4
*x^4 - 10*e^5*x^5) - 3465*d^2*(d - e*x)^3*ArcSin[(e*x)/d]))/(30*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^
2*x^2)/d^2])

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IntegrateAlgebraic [A]  time = 0.59, size = 134, normalized size = 0.65 \begin {gather*} -\frac {231 d^3 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{2 e^2}-\frac {\sqrt {d^2-e^2 x^2} \left (5446 d^5-12843 d^4 e x+8711 d^3 e^2 x^2-815 d^2 e^3 x^3-105 d e^4 x^4-10 e^5 x^5\right )}{30 e (e x-d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

-1/30*(Sqrt[d^2 - e^2*x^2]*(5446*d^5 - 12843*d^4*e*x + 8711*d^3*e^2*x^2 - 815*d^2*e^3*x^3 - 105*d*e^4*x^4 - 10
*e^5*x^5))/(e*(-d + e*x)^3) - (231*d^3*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*e^2)

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fricas [A]  time = 0.48, size = 201, normalized size = 0.98 \begin {gather*} \frac {5446 \, d^{3} e^{3} x^{3} - 16338 \, d^{4} e^{2} x^{2} + 16338 \, d^{5} e x - 5446 \, d^{6} + 6930 \, {\left (d^{3} e^{3} x^{3} - 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x - d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (10 \, e^{5} x^{5} + 105 \, d e^{4} x^{4} + 815 \, d^{2} e^{3} x^{3} - 8711 \, d^{3} e^{2} x^{2} + 12843 \, d^{4} e x - 5446 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(5446*d^3*e^3*x^3 - 16338*d^4*e^2*x^2 + 16338*d^5*e*x - 5446*d^6 + 6930*(d^3*e^3*x^3 - 3*d^4*e^2*x^2 + 3*
d^5*e*x - d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (10*e^5*x^5 + 105*d*e^4*x^4 + 815*d^2*e^3*x^3 - 871
1*d^3*e^2*x^2 + 12843*d^4*e*x - 5446*d^5)*sqrt(-e^2*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 + 3*d^2*e^2*x - d^3*e)

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giac [A]  time = 0.37, size = 129, normalized size = 0.63 \begin {gather*} -\frac {231}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {{\left (5446 \, d^{8} e^{\left (-1\right )} + {\left (3495 \, d^{7} - {\left (13480 \, d^{6} e + {\left (7765 \, d^{5} e^{2} - {\left (10740 \, d^{4} e^{3} + {\left (5941 \, d^{3} e^{4} - 5 \, {\left (232 \, d^{2} e^{5} + {\left (2 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-231/2*d^3*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/30*(5446*d^8*e^(-1) + (3495*d^7 - (13480*d^6*e + (7765*d^5*e^2 - (1
0740*d^4*e^3 + (5941*d^3*e^4 - 5*(232*d^2*e^5 + (2*x*e^7 + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)
/(x^2*e^2 - d^2)^3

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maple [A]  time = 0.29, size = 309, normalized size = 1.50 \begin {gather*} -\frac {e^{7} x^{8}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {9 d \,e^{6} x^{7}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {116 d^{2} e^{5} x^{6}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {231 d^{3} e^{4} x^{5}}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {358 d^{4} e^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {63 d^{5} e^{2} x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {1348 d^{6} e \,x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {152 d^{7} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {77 d^{3} e^{2} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2723 d^{8}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {157 d^{5} x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4093 d^{3} x}{30 \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {231 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x)

[Out]

63*d^5*e^2*x^3/(-e^2*x^2+d^2)^(5/2)-9/2*d*e^6*x^7/(-e^2*x^2+d^2)^(5/2)+231/10*d^3*e^4*x^5/(-e^2*x^2+d^2)^(5/2)
-77/2*d^3*e^2*x^3/(-e^2*x^2+d^2)^(3/2)-116/3*e^5*d^2*x^6/(-e^2*x^2+d^2)^(5/2)+358*e^3*d^4*x^4/(-e^2*x^2+d^2)^(
5/2)-1348/3*e*d^6*x^2/(-e^2*x^2+d^2)^(5/2)-231/2/(e^2)^(1/2)*d^3*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)-15
2/5*d^7*x/(-e^2*x^2+d^2)^(5/2)+157/15*d^5*x/(-e^2*x^2+d^2)^(3/2)+2723/15*d^8/e/(-e^2*x^2+d^2)^(5/2)+4093/30*d^
3*x/(-e^2*x^2+d^2)^(1/2)-1/3*e^7*x^8/(-e^2*x^2+d^2)^(5/2)

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maxima [B]  time = 3.17, size = 374, normalized size = 1.82 \begin {gather*} -\frac {e^{7} x^{8}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {9 \, d e^{6} x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {77}{10} \, d^{3} e^{6} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {116 \, d^{2} e^{5} x^{6}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {77}{2} \, d^{3} e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {358 \, d^{4} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {63 \, d^{5} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {1348 \, d^{6} e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {152 \, d^{7} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2723 \, d^{8}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {619 \, d^{5} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {989 \, d^{3} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {231 \, d^{3} \arcsin \left (\frac {e x}{d}\right )}{2 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/3*e^7*x^8/(-e^2*x^2 + d^2)^(5/2) - 9/2*d*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 77/10*d^3*e^6*x*(15*x^4/((-e^2*x^
2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 116/3*d^
2*e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 77/2*d^3*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2
)^(3/2)*e^4)) + 358*d^4*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 63*d^5*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 1348/3*d^6*e*
x^2/(-e^2*x^2 + d^2)^(5/2) - 152/5*d^7*x/(-e^2*x^2 + d^2)^(5/2) + 2723/15*d^8/((-e^2*x^2 + d^2)^(5/2)*e) + 619
/15*d^5*x/(-e^2*x^2 + d^2)^(3/2) - 989/30*d^3*x/sqrt(-e^2*x^2 + d^2) - 231/2*d^3*arcsin(e*x/d)/e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^9}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int((d + e*x)^9/(d^2 - e^2*x^2)^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{9}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**9/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**9/(-(-d + e*x)*(d + e*x))**(7/2), x)

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