3.1635 \(\int \frac{(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=137 \[ -\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{7 e \sqrt{d+e x} (b d-a e)^2}{b^4}+\frac{7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{7 e (d+e x)^{5/2}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.224456, antiderivative size = 137, 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 \[ -\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{7 e \sqrt{d+e x} (b d-a e)^2}{b^4}+\frac{7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{7 e (d+e x)^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.1277, size = 122, normalized size = 0.89 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{b \left (a + b x\right )} + \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{5 b^{2}} - \frac{7 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{3 b^{3}} + \frac{7 e \sqrt{d + e x} \left (a e - b d\right )^{2}}{b^{4}} - \frac{7 e \left (a e - b d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.403615, size = 138, normalized size = 1.01 \[ \frac{\sqrt{d+e x} \left (2 e \left (45 a^2 e^2-100 a b d e+58 b^2 d^2\right )+4 b e^2 x (8 b d-5 a e)-\frac{15 (b d-a e)^3}{a+b x}+6 b^2 e^3 x^2\right )}{15 b^4}-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(2*e*(58*b^2*d^2 - 100*a*b*d*e + 45*a^2*e^2) + 4*b*e^2*(8*b*d - 5
*a*e)*x + 6*b^2*e^3*x^2 - (15*(b*d - a*e)^3)/(a + b*x)))/(15*b^4) - (7*e*(b*d -
a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)

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Maple [B]  time = 0.024, size = 387, normalized size = 2.8 \[{\frac{2\,e}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{4\,a{e}^{2}}{3\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,de}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}{e}^{3}\sqrt{ex+d}}{{b}^{4}}}-12\,{\frac{ad{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+6\,{\frac{e{d}^{2}\sqrt{ex+d}}{{b}^{2}}}+{\frac{{a}^{3}{e}^{4}}{{b}^{4} \left ( bex+ae \right ) }\sqrt{ex+d}}-3\,{\frac{\sqrt{ex+d}{a}^{2}d{e}^{3}}{{b}^{3} \left ( bex+ae \right ) }}+3\,{\frac{\sqrt{ex+d}a{d}^{2}{e}^{2}}{{b}^{2} \left ( bex+ae \right ) }}-{\frac{e{d}^{3}}{b \left ( bex+ae \right ) }\sqrt{ex+d}}-7\,{\frac{{a}^{3}{e}^{4}}{{b}^{4}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+21\,{\frac{{a}^{2}d{e}^{3}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-21\,{\frac{a{d}^{2}{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+7\,{\frac{e{d}^{3}}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*e*(e*x+d)^(5/2)/b^2-4/3/b^3*a*e^2*(e*x+d)^(3/2)+4/3*e/b^2*d*(e*x+d)^(3/2)+6/
b^4*a^2*e^3*(e*x+d)^(1/2)-12/b^3*d*e^2*a*(e*x+d)^(1/2)+6*e/b^2*d^2*(e*x+d)^(1/2)
+1/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*a^3*e^4-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*a^2*d*e
^3+3/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*a*d^2*e^2-e/b*(e*x+d)^(1/2)/(b*e*x+a*e)*d^3-7
/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*e^4+21/
b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*d*e^3-21
/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*d^2*e^2+7
*e/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22127, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/30*(105*(a*b^2*d^2*e - 2*a^2*b*d*e^2 + a^3*e^3 + (b^3*d^2*e - 2*a*b^2*d*e^2 +
a^2*b*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*
sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(6*b^3*e^3*x^3 - 15*b^3*d^3 + 161*a*b^2*d^2*
e - 245*a^2*b*d*e^2 + 105*a^3*e^3 + 2*(16*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(58*b
^3*d^2*e - 84*a*b^2*d*e^2 + 35*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^5*x + a*b^4), -1/
15*(105*(a*b^2*d^2*e - 2*a^2*b*d*e^2 + a^3*e^3 + (b^3*d^2*e - 2*a*b^2*d*e^2 + a^
2*b*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (6
*b^3*e^3*x^3 - 15*b^3*d^3 + 161*a*b^2*d^2*e - 245*a^2*b*d*e^2 + 105*a^3*e^3 + 2*
(16*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(58*b^3*d^2*e - 84*a*b^2*d*e^2 + 35*a^2*b*e
^3)*x)*sqrt(e*x + d))/(b^5*x + a*b^4)]

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

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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21766, size = 379, normalized size = 2.77 \[ \frac{7 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} - \frac{\sqrt{x e + d} b^{3} d^{3} e - 3 \, \sqrt{x e + d} a b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} a^{2} b d e^{3} - \sqrt{x e + d} a^{3} e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{8} e + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{8} d e + 45 \, \sqrt{x e + d} b^{8} d^{2} e - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{7} e^{2} - 90 \, \sqrt{x e + d} a b^{7} d e^{2} + 45 \, \sqrt{x e + d} a^{2} b^{6} e^{3}\right )}}{15 \, b^{10}} \]

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),x, algorithm="giac")

[Out]

7*(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*arctan(sqrt(x*e + d)*b
/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) - (sqrt(x*e + d)*b^3*d^3*e - 3
*sqrt(x*e + d)*a*b^2*d^2*e^2 + 3*sqrt(x*e + d)*a^2*b*d*e^3 - sqrt(x*e + d)*a^3*e
^4)/(((x*e + d)*b - b*d + a*e)*b^4) + 2/15*(3*(x*e + d)^(5/2)*b^8*e + 10*(x*e +
d)^(3/2)*b^8*d*e + 45*sqrt(x*e + d)*b^8*d^2*e - 10*(x*e + d)^(3/2)*a*b^7*e^2 - 9
0*sqrt(x*e + d)*a*b^7*d*e^2 + 45*sqrt(x*e + d)*a^2*b^6*e^3)/b^10