3.1667 \(\int \frac{(d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=178 \[ -\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x
)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5)
- (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

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Rubi [A]  time = 0.0901336, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 47, 63, 208} \[ -\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x
)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5)
- (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{9/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{\left (63 e^2\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{\left (21 e^3\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{\left (63 e^4\right ) \int \frac{\sqrt{d+e x}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{\left (63 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^5}\\ &=-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac{\left (63 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 b^5}\\ &=-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}\\ \end{align*}

Mathematica [A]  time = 0.314815, size = 178, normalized size = 1. \[ -\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}+\frac{63 e^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{128 b^{11/2} \sqrt{a e-b d}}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x
)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5)
+ (63*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*Sqrt[-(b*d) + a*e])

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Maple [B]  time = 0.209, size = 463, normalized size = 2.6 \begin{align*} -{\frac{193\,{e}^{5}}{128\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{9}{2}}}}-{\frac{237\,{e}^{6}a}{64\, \left ( bxe+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{237\,{e}^{5}d}{64\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{21\,{a}^{2}{e}^{7}}{5\, \left ( bxe+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{42\,{e}^{6}ad}{5\, \left ( bxe+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{21\,{e}^{5}{d}^{2}}{5\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{147\,{e}^{8}{a}^{3}}{64\, \left ( bxe+ae \right ) ^{5}{b}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{441\,{e}^{7}d{a}^{2}}{64\, \left ( bxe+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{441\,{e}^{6}a{d}^{2}}{64\, \left ( bxe+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{147\,{e}^{5}{d}^{3}}{64\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{63\,{e}^{9}{a}^{4}}{128\, \left ( bxe+ae \right ) ^{5}{b}^{5}}\sqrt{ex+d}}+{\frac{63\,{e}^{8}{a}^{3}d}{32\, \left ( bxe+ae \right ) ^{5}{b}^{4}}\sqrt{ex+d}}-{\frac{189\,{e}^{7}{d}^{2}{a}^{2}}{64\, \left ( bxe+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}+{\frac{63\,{e}^{6}a{d}^{3}}{32\, \left ( bxe+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}-{\frac{63\,{e}^{5}{d}^{4}}{128\, \left ( bxe+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-193/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(9/2)-237/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(7/2)*a+237/64*e^5/(b*e*x+a*e)
^5/b*(e*x+d)^(7/2)*d-21/5*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(5/2)*a^2+42/5*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*a*d
-21/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*d^2-147/64*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(3/2)*a^3+441/64*e^7/(b*e*x+a
*e)^5/b^3*(e*x+d)^(3/2)*d*a^2-441/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*a*d^2+147/64*e^5/(b*e*x+a*e)^5/b*(e*x
+d)^(3/2)*d^3-63/128*e^9/(b*e*x+a*e)^5/b^5*(e*x+d)^(1/2)*a^4+63/32*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*a^3*d-1
89/64*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*d^2*a^2+63/32*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*a*d^3-63/128*e^5/(
b*e*x+a*e)^5/b*(e*x+d)^(1/2)*d^4+63/128*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)
)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14142, size = 2122, normalized size = 11.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^
5)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(128*b^6
*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6
*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2 + 6
1*a*b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*b^4*
d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11*e)*
x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3*b^9*d - a^4*b^8*e)*x^2 + 5*(a^
4*b^8*d - a^5*b^7*e)*x), 1/640*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 +
5*a^4*b*e^5*x + a^5*e^5)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (128
*b^6*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*
(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2
+ 61*a*b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*
b^4*d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11
*e)*x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3*b^9*d - a^4*b^8*e)*x^2 + 5
*(a^4*b^8*d - a^5*b^7*e)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.26686, size = 451, normalized size = 2.53 \begin{align*} \frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{965 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 1260 \, \sqrt{x e + d} a^{3} b d e^{8} + 315 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

63/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/(sqrt(-b^2*d + a*b*e)*b^5) - 1/640*(965*(x*e + d)^(9/2
)*b^4*e^5 - 2370*(x*e + d)^(7/2)*b^4*d*e^5 + 2688*(x*e + d)^(5/2)*b^4*d^2*e^5 - 1470*(x*e + d)^(3/2)*b^4*d^3*e
^5 + 315*sqrt(x*e + d)*b^4*d^4*e^5 + 2370*(x*e + d)^(7/2)*a*b^3*e^6 - 5376*(x*e + d)^(5/2)*a*b^3*d*e^6 + 4410*
(x*e + d)^(3/2)*a*b^3*d^2*e^6 - 1260*sqrt(x*e + d)*a*b^3*d^3*e^6 + 2688*(x*e + d)^(5/2)*a^2*b^2*e^7 - 4410*(x*
e + d)^(3/2)*a^2*b^2*d*e^7 + 1890*sqrt(x*e + d)*a^2*b^2*d^2*e^7 + 1470*(x*e + d)^(3/2)*a^3*b*e^8 - 1260*sqrt(x
*e + d)*a^3*b*d*e^8 + 315*sqrt(x*e + d)*a^4*e^9)/(((x*e + d)*b - b*d + a*e)^5*b^5)