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

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

[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*ArcTanh[(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 = 0.243236, 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 \[ -\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 (a+b x) (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x)^2 (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^4 (b d-a e)} \]

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*ArcTanh[(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 in Sympy [A]  time = 77.9962, size = 158, normalized size = 0.88 \[ \frac{35 e^{3} \sqrt{d + e x}}{64 \left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{35 e^{2} \sqrt{d + e x}}{96 \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} + \frac{7 e \sqrt{d + e x}}{24 \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x}}{4 \left (a + b x\right )^{4} \left (a e - b d\right )} + \frac{35 e^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{64 \sqrt{b} \left (a e - b d\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

35*e**3*sqrt(d + e*x)/(64*(a + b*x)*(a*e - b*d)**4) + 35*e**2*sqrt(d + e*x)/(96*
(a + b*x)**2*(a*e - b*d)**3) + 7*e*sqrt(d + e*x)/(24*(a + b*x)**3*(a*e - b*d)**2
) + sqrt(d + e*x)/(4*(a + b*x)**4*(a*e - b*d)) + 35*e**4*atan(sqrt(b)*sqrt(d + e
*x)/sqrt(a*e - b*d))/(64*sqrt(b)*(a*e - b*d)**(9/2))

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Mathematica [A]  time = 0.522148, size = 145, normalized size = 0.81 \[ \frac{1}{192} \left (\frac{\sqrt{d+e x} \left (70 e^2 (a+b x)^2 (a e-b d)+56 e (a+b x) (b d-a e)^2-48 (b d-a e)^3+105 e^3 (a+b x)^3\right )}{(a+b x)^4 (b d-a e)^4}-\frac{105 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{9/2}}\right ) \]

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]

((Sqrt[d + e*x]*(-48*(b*d - a*e)^3 + 56*e*(b*d - a*e)^2*(a + b*x) + 70*e^2*(-(b*
d) + a*e)*(a + b*x)^2 + 105*e^3*(a + b*x)^3))/((b*d - a*e)^4*(a + b*x)^4) - (105
*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(9/2
)))/192

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Maple [A]  time = 0.013, size = 179, normalized size = 1. \[{\frac{{e}^{4}}{ \left ( 4\,ae-4\,bd \right ) \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{7\,{e}^{4}}{24\, \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{96\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{64\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{35\,{e}^{4}}{64\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

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/(b*(a*e-b*d))^(1/2)*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307287, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/384*(2*(105*b^3*e^3*x^3 - 48*b^3*d^3 + 200*a*b^2*d^2*e - 326*a^2*b*d*e^2 + 27
9*a^3*e^3 - 35*(2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 + 7*(8*b^3*d^2*e - 36*a*b^2*d*e^
2 + 73*a^2*b*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 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)*log((sqrt(b^2*d - a*b
*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^4*b
^4*d^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2 - 4*a^7*b*d*e^3 + a^8*e^4 + (b^8*d^
4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4)*x^4 + 4*(
a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3*b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a^5*b^3*e^4)
*x^3 + 6*(a^2*b^6*d^4 - 4*a^3*b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b^3*d*e^3 +
a^6*b^2*e^4)*x^2 + 4*(a^3*b^5*d^4 - 4*a^4*b^4*d^3*e + 6*a^5*b^3*d^2*e^2 - 4*a^6*
b^2*d*e^3 + a^7*b*e^4)*x)*sqrt(b^2*d - a*b*e)), 1/192*((105*b^3*e^3*x^3 - 48*b^3
*d^3 + 200*a*b^2*d^2*e - 326*a^2*b*d*e^2 + 279*a^3*e^3 - 35*(2*b^3*d*e^2 - 11*a*
b^2*e^3)*x^2 + 7*(8*b^3*d^2*e - 36*a*b^2*d*e^2 + 73*a^2*b*e^3)*x)*sqrt(-b^2*d +
a*b*e)*sqrt(e*x + d) - 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)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)
)))/((a^4*b^4*d^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2 - 4*a^7*b*d*e^3 + a^8*e^
4 + (b^8*d^4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4
)*x^4 + 4*(a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3*b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a
^5*b^3*e^4)*x^3 + 6*(a^2*b^6*d^4 - 4*a^3*b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b
^3*d*e^3 + a^6*b^2*e^4)*x^2 + 4*(a^3*b^5*d^4 - 4*a^4*b^4*d^3*e + 6*a^5*b^3*d^2*e
^2 - 4*a^6*b^2*d*e^3 + a^7*b*e^4)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{5} \sqrt{d + e x}}\, dx \]

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]

Integral(1/((a + b*x)**5*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.292432, size = 447, normalized size = 2.48 \[ \frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{105 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 385 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 1022 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 837 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 837 \, \sqrt{x e + d} a^{2} b d e^{6} + 279 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

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*sqrt(e*x + d)),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*(1
05*(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 - 1
022*(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)