3.1642 \(\int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=151 \[ \frac{7 b^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac{7 b^2 e}{\sqrt{d+e x} (b d-a e)^4}-\frac{7 b e}{3 (d+e x)^{3/2} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{5/2} (b d-a e)}-\frac{7 e}{5 (d+e x)^{5/2} (b d-a e)^2} \]

[Out]

(-7*e)/(5*(b*d - a*e)^2*(d + e*x)^(5/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(5
/2)) - (7*b*e)/(3*(b*d - a*e)^3*(d + e*x)^(3/2)) - (7*b^2*e)/((b*d - a*e)^4*Sqrt
[d + e*x]) + (7*b^(5/2)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d
- a*e)^(9/2)

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Rubi [A]  time = 0.306437, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 b^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac{7 b^2 e}{\sqrt{d+e x} (b d-a e)^4}-\frac{7 b e}{3 (d+e x)^{3/2} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{5/2} (b d-a e)}-\frac{7 e}{5 (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-7*e)/(5*(b*d - a*e)^2*(d + e*x)^(5/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(5
/2)) - (7*b*e)/(3*(b*d - a*e)^3*(d + e*x)^(3/2)) - (7*b^2*e)/((b*d - a*e)^4*Sqrt
[d + e*x]) + (7*b^(5/2)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d
- a*e)^(9/2)

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Rubi in Sympy [A]  time = 64.2469, size = 134, normalized size = 0.89 \[ - \frac{7 b^{\frac{5}{2}} e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{9}{2}}} - \frac{7 b^{2} e}{\sqrt{d + e x} \left (a e - b d\right )^{4}} + \frac{7 b e}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} - \frac{7 e}{5 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-7*b**(5/2)*e*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(a*e - b*d)**(9/2) - 7
*b**2*e/(sqrt(d + e*x)*(a*e - b*d)**4) + 7*b*e/(3*(d + e*x)**(3/2)*(a*e - b*d)**
3) - 7*e/(5*(d + e*x)**(5/2)*(a*e - b*d)**2) + 1/((a + b*x)*(d + e*x)**(5/2)*(a*
e - b*d))

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Mathematica [A]  time = 0.542932, size = 137, normalized size = 0.91 \[ \frac{7 b^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (\frac{15 b^3}{a+b x}+\frac{20 b e (b d-a e)}{(d+e x)^2}+\frac{6 e (b d-a e)^2}{(d+e x)^3}+\frac{90 b^2 e}{d+e x}\right )}{15 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

-(Sqrt[d + e*x]*((15*b^3)/(a + b*x) + (6*e*(b*d - a*e)^2)/(d + e*x)^3 + (20*b*e*
(b*d - a*e))/(d + e*x)^2 + (90*b^2*e)/(d + e*x)))/(15*(b*d - a*e)^4) + (7*b^(5/2
)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(9/2)

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Maple [A]  time = 0.028, size = 149, normalized size = 1. \[ -{\frac{2\,e}{5\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-6\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{4\,be}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{{b}^{3}e}{ \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) }\sqrt{ex+d}}-7\,{\frac{{b}^{3}e}{ \left ( ae-bd \right ) ^{4}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-2/5*e/(a*e-b*d)^2/(e*x+d)^(5/2)-6*e/(a*e-b*d)^4*b^2/(e*x+d)^(1/2)+4/3*e/(a*e-b*
d)^3*b/(e*x+d)^(3/2)-e*b^3/(a*e-b*d)^4*(e*x+d)^(1/2)/(b*e*x+a*e)-7*e*b^3/(a*e-b*
d)^4/(b*(a*e-b*d))^(1/2)*arctan((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(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230973, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/30*(210*b^3*e^3*x^3 + 30*b^3*d^3 + 232*a*b^2*d^2*e - 64*a^2*b*d*e^2 + 12*a^3
*e^3 + 70*(7*b^3*d*e^2 + 2*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*
b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b^2*d*e^2)*x)*sqrt(e*x + d)*sqrt(b
/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d
- a*e)))/(b*x + a)) + 14*(23*b^3*d^2*e + 24*a*b^2*d*e^2 - 2*a^2*b*e^3)*x)/((a*b
^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (
b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)
*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2 + 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 -
2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 - 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*
a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5*d*e^5)*x)*sqrt(e*x + d)), -1/15*(105*b
^3*e^3*x^3 + 15*b^3*d^3 + 116*a*b^2*d^2*e - 32*a^2*b*d*e^2 + 6*a^3*e^3 + 35*(7*b
^3*d*e^2 + 2*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*b^3*d*e^2 + a*
b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b^2*d*e^2)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) + 7*(23*b^3*d^2*e +
24*a*b^2*d*e^2 - 2*a^2*b*e^3)*x)/((a*b^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*
e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3
*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2 + 8
*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 -
2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5
*d*e^5)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Integral(1/((a + b*x)**2*(d + e*x)**(7/2)), x)

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GIAC/XCAS [A]  time = 0.21342, size = 410, normalized size = 2.72 \[ -\frac{7 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{\sqrt{x e + d} b^{3} e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} b^{2} e + 10 \,{\left (x e + d\right )} b^{2} d e + 3 \, b^{2} d^{2} e - 10 \,{\left (x e + d\right )} a b e^{2} - 6 \, a b d e^{2} + 3 \, a^{2} e^{3}\right )}}{15 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

-7*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e/((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)) - sqrt(x*e
+ d)*b^3*e/((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)) - 2/15*(45*(x*e + d)^2*b^2*e + 10*(x*e + d)*b^2*d
*e + 3*b^2*d^2*e - 10*(x*e + d)*a*b*e^2 - 6*a*b*d*e^2 + 3*a^2*e^3)/((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)^(5/2))