3.1668 \(\int \frac{(d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=188 \[ -\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (a+b x) (b d-a e)}-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}+\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5} \]

[Out]

(-7*e^3*Sqrt[d + e*x])/(64*b^4*(a + b*x)^2) - (7*e^4*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)*(a + b*x)) - (7*e^2*(
d + e*x)^(3/2))/(48*b^3*(a + b*x)^3) - (7*e*(d + e*x)^(5/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(7/2)/(5*b*(a +
b*x)^5) + (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(3/2))

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Rubi [A]  time = 0.10614, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 51, 63, 208} \[ -\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (a+b x) (b d-a e)}-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}+\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-7*e^3*Sqrt[d + e*x])/(64*b^4*(a + b*x)^2) - (7*e^4*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)*(a + b*x)) - (7*e^2*(
d + e*x)^(3/2))/(48*b^3*(a + b*x)^3) - (7*e*(d + e*x)^(5/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(7/2)/(5*b*(a +
b*x)^5) + (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(3/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{7/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac{(7 e) \int \frac{(d+e x)^{5/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac{\left (7 e^2\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac{\left (7 e^3\right ) \int \frac{\sqrt{d+e x}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac{\left (7 e^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b^4}\\ &=-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (b d-a e) (a+b x)}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}-\frac{\left (7 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^4 (b d-a e)}\\ &=-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (b d-a e) (a+b x)}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}-\frac{\left (7 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^4 (b d-a e)}\\ &=-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (b d-a e) (a+b x)}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.019642, size = 52, normalized size = 0.28 \[ \frac{2 e^5 (d+e x)^{9/2} \, _2F_1\left (\frac{9}{2},6;\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right )}{9 (a e-b d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(9/2)*Hypergeometric2F1[9/2, 6, 11/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(9*(-(b*d) + a*e)^6)

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Maple [B]  time = 0.258, size = 360, normalized size = 1.9 \begin{align*}{\frac{7\,{e}^{5}}{128\, \left ( bxe+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}-{\frac{79\,{e}^{5}}{192\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{e}^{6}a}{15\, \left ( bxe+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{e}^{5}d}{15\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{49\,{a}^{2}{e}^{7}}{192\, \left ( bxe+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{49\,{e}^{6}ad}{96\, \left ( bxe+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{49\,{e}^{5}{d}^{2}}{192\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{8}{a}^{3}}{128\, \left ( bxe+ae \right ) ^{5}{b}^{4}}\sqrt{ex+d}}+{\frac{21\,{e}^{7}d{a}^{2}}{128\, \left ( bxe+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}-{\frac{21\,{e}^{6}a{d}^{2}}{128\, \left ( bxe+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}+{\frac{7\,{e}^{5}{d}^{3}}{128\, \left ( bxe+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{7\,{e}^{5}}{ \left ( 128\,ae-128\,bd \right ){b}^{4}}\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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

7/128*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)-79/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(7/2)-7/15*e^6/(b*e*x+a*e)^
5/b^2*(e*x+d)^(5/2)*a+7/15*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*d-49/192*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*a^2+
49/96*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*a*d-49/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*d^2-7/128*e^8/(b*e*x+a*
e)^5/b^4*(e*x+d)^(1/2)*a^3+21/128*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*d*a^2-21/128*e^6/(b*e*x+a*e)^5/b^2*(e*x+
d)^(1/2)*a*d^2+7/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*d^3+7/128*e^5/(a*e-b*d)/b^4/((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)^(7/2)/(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 = 1.898, size = 2458, normalized size = 13.07 \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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(105*(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*(384*b^
6*d^5 - 432*a*b^5*d^4*e - 8*a^2*b^4*d^3*e^2 - 14*a^3*b^3*d^2*e^3 - 35*a^4*b^2*d*e^4 + 105*a^5*b*e^5 + 105*(b^6
*d*e^4 - a*b^5*e^5)*x^4 + 10*(121*b^6*d^2*e^3 - 200*a*b^5*d*e^4 + 79*a^2*b^4*e^5)*x^3 + 2*(1052*b^6*d^3*e^2 -
1341*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*(744*b^6*d^4*e - 872*a*b^5*d^3*e^2 - 33*a^2*
b^4*d^2*e^3 - 84*a^3*b^3*d*e^4 + 245*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^2 - 2*a^6*b^6*d*e + a^7*b^5*e^2
+ (b^12*d^2 - 2*a*b^11*d*e + a^2*b^10*e^2)*x^5 + 5*(a*b^11*d^2 - 2*a^2*b^10*d*e + a^3*b^9*e^2)*x^4 + 10*(a^2*
b^10*d^2 - 2*a^3*b^9*d*e + a^4*b^8*e^2)*x^3 + 10*(a^3*b^9*d^2 - 2*a^4*b^8*d*e + a^5*b^7*e^2)*x^2 + 5*(a^4*b^8*
d^2 - 2*a^5*b^7*d*e + a^6*b^6*e^2)*x), -1/1920*(105*(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)) + (384*b^6*d^5 - 432*a*b^5*d^4*e - 8*a^2*b^4*d^3*e^2 - 14*a^3*b^3*d^2*e^3 - 35*a^4*b^2*d*e^4 + 105*a
^5*b*e^5 + 105*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(121*b^6*d^2*e^3 - 200*a*b^5*d*e^4 + 79*a^2*b^4*e^5)*x^3 + 2*(
1052*b^6*d^3*e^2 - 1341*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*(744*b^6*d^4*e - 872*a*b^
5*d^3*e^2 - 33*a^2*b^4*d^2*e^3 - 84*a^3*b^3*d*e^4 + 245*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^2 - 2*a^6*b^
6*d*e + a^7*b^5*e^2 + (b^12*d^2 - 2*a*b^11*d*e + a^2*b^10*e^2)*x^5 + 5*(a*b^11*d^2 - 2*a^2*b^10*d*e + a^3*b^9*
e^2)*x^4 + 10*(a^2*b^10*d^2 - 2*a^3*b^9*d*e + a^4*b^8*e^2)*x^3 + 10*(a^3*b^9*d^2 - 2*a^4*b^8*d*e + a^5*b^7*e^2
)*x^2 + 5*(a^4*b^8*d^2 - 2*a^5*b^7*d*e + a^6*b^6*e^2)*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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.26487, size = 486, normalized size = 2.59 \begin{align*} -\frac{7 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d - a b^{4} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{105 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} + 790 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} - 896 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} + 490 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt{x e + d} b^{4} d^{4} e^{5} - 790 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} + 1792 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} - 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} - 896 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} + 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} - 490 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 420 \, \sqrt{x e + d} a^{3} b d e^{8} - 105 \, \sqrt{x e + d} a^{4} e^{9}}{1920 \,{\left (b^{5} d - a b^{4} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-7/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d - a*b^4*e)*sqrt(-b^2*d + a*b*e)) - 1/1920*(105
*(x*e + d)^(9/2)*b^4*e^5 + 790*(x*e + d)^(7/2)*b^4*d*e^5 - 896*(x*e + d)^(5/2)*b^4*d^2*e^5 + 490*(x*e + d)^(3/
2)*b^4*d^3*e^5 - 105*sqrt(x*e + d)*b^4*d^4*e^5 - 790*(x*e + d)^(7/2)*a*b^3*e^6 + 1792*(x*e + d)^(5/2)*a*b^3*d*
e^6 - 1470*(x*e + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(x*e + d)*a*b^3*d^3*e^6 - 896*(x*e + d)^(5/2)*a^2*b^2*e^7 +
1470*(x*e + d)^(3/2)*a^2*b^2*d*e^7 - 630*sqrt(x*e + d)*a^2*b^2*d^2*e^7 - 490*(x*e + d)^(3/2)*a^3*b*e^8 + 420*
sqrt(x*e + d)*a^3*b*d*e^8 - 105*sqrt(x*e + d)*a^4*e^9)/((b^5*d - a*b^4*e)*((x*e + d)*b - b*d + a*e)^5)