3.2089 \(\int \frac{a+b x}{\sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac{35 e^3 \sqrt{d+e x}}{64 (a+b x) (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x)^2 (b d-a e)^3}-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} (b d-a e)^{9/2}}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^4 (b d-a e)} \]

[Out]

-Sqrt[d + e*x]/(4*(b*d - a*e)*(a + b*x)^4) + (7*e*Sqrt[d + e*x])/(24*(b*d - a*e)^2*(a + b*x)^3) - (35*e^2*Sqrt
[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)^2) + (35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)) - (35*e^4*ArcT
anh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.0949907, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \[ \frac{35 e^3 \sqrt{d+e x}}{64 (a+b x) (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x)^2 (b d-a e)^3}-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} (b d-a e)^{9/2}}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[d + e*x]/(4*(b*d - a*e)*(a + b*x)^4) + (7*e*Sqrt[d + e*x])/(24*(b*d - a*e)^2*(a + b*x)^3) - (35*e^2*Sqrt
[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)^2) + (35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)) - (35*e^4*ArcT
anh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(9/2))

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 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{a+b x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}-\frac{(7 e) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{8 (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^3}+\frac{\left (35 e^2\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{48 (b d-a e)^2}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x)^2}-\frac{\left (35 e^3\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{64 (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)}+\frac{\left (35 e^4\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{128 (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)}+\frac{\left (35 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{64 (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{4 (b d-a e) (a+b x)^4}+\frac{7 e \sqrt{d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac{35 e^2 \sqrt{d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac{35 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)}-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} (b d-a e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0118434, size = 50, normalized size = 0.28 \[ \frac{2 e^4 \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{(a e-b d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e^4*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 5, 3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(-(b*d) + a*e)^5

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Maple [A]  time = 0.01, size = 179, normalized size = 1. \begin{align*}{\frac{{e}^{4}}{ \left ( 4\,ae-4\,bd \right ) \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{7\,{e}^{4}}{24\, \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{96\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{64\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{64\, \left ( ae-bd \right ) ^{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((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

1/4*e^4*(e*x+d)^(1/2)/(a*e-b*d)/(b*e*x+a*e)^4+7/24*e^4/(a*e-b*d)^2*(e*x+d)^(1/2)/(b*e*x+a*e)^3+35/96*e^4/(a*e-
b*d)^3*(e*x+d)^(1/2)/(b*e*x+a*e)^2+35/64*e^4/(a*e-b*d)^4*(e*x+d)^(1/2)/(b*e*x+a*e)+35/64*e^4/(a*e-b*d)^4/((a*e
-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((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((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.22284, size = 2709, normalized size = 15.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/384*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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*(48*b^5*d^4 - 248*a*b^4*d^3*e +
526*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279*a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e^2
- 13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2*e
^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e
^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b
^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6
*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 -
a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e
^4 - a^8*b^2*e^5)*x), 1/192*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)
*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (48*b^5*d^4 - 248*a*b^4*d^3*e
+ 526*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279*a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e
^2 - 13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2
*e^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d
*e^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5
*b^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a
^6*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4
- a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d
*e^4 - a^8*b^2*e^5)*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((b*x+a)/(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23175, size = 447, normalized size = 2.48 \begin{align*} \frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{105 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 385 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 1022 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 837 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 837 \, \sqrt{x e + d} a^{2} b d e^{6} + 279 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

35/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b
*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) + 1/192*(105*(x*e + d)^(7/2)*b^3*e^4 - 385*(x*e + d)^(5/2)*b^3*d*e^4 +
511*(x*e + d)^(3/2)*b^3*d^2*e^4 - 279*sqrt(x*e + d)*b^3*d^3*e^4 + 385*(x*e + d)^(5/2)*a*b^2*e^5 - 1022*(x*e +
d)^(3/2)*a*b^2*d*e^5 + 837*sqrt(x*e + d)*a*b^2*d^2*e^5 + 511*(x*e + d)^(3/2)*a^2*b*e^6 - 837*sqrt(x*e + d)*a^
2*b*d*e^6 + 279*sqrt(x*e + d)*a^3*e^7)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4
)*((x*e + d)*b - b*d + a*e)^4)