3.2066 \(\int \frac{(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{2 (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{2 \sqrt{d+e x} (b d-a e)^3}{b^4}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 b^3}+\frac{2 (d+e x)^{5/2} (b d-a e)}{5 b^2}+\frac{2 (d+e x)^{7/2}}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.28564, antiderivative size = 138, 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 \[ -\frac{2 (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{2 \sqrt{d+e x} (b d-a e)^3}{b^4}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 b^3}+\frac{2 (d+e x)^{5/2} (b d-a e)}{5 b^2}+\frac{2 (d+e x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.5559, size = 122, normalized size = 0.88 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}}}{7 b} - \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{5 b^{2}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{3 b^{3}} - \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{3}}{b^{4}} + \frac{2 \left (a e - b d\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.211747, size = 149, normalized size = 1.08 \[ \frac{2 \sqrt{d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 b^4}-\frac{2 (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(-105*a^3*e^3 + 35*a^2*b*e^2*(10*d + e*x) - 7*a*b^2*e*(58*d^2 +
16*d*e*x + 3*e^2*x^2) + b^3*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3)
))/(105*b^4) - (2*(b*d - a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a
*e]])/b^(9/2)

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Maple [B]  time = 0.085, size = 380, normalized size = 2.8 \[{\frac{2}{7\,b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{2\,ae}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,d}{5\,b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}{e}^{2}}{3\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{4\,aed}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{d}^{2}}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}{e}^{3}\sqrt{ex+d}}{{b}^{4}}}+6\,{\frac{{a}^{2}d{e}^{2}\sqrt{ex+d}}{{b}^{3}}}-6\,{\frac{a{d}^{2}e\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{{d}^{3}\sqrt{ex+d}}{b}}+2\,{\frac{{a}^{4}{e}^{4}}{{b}^{4}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-8\,{\frac{{a}^{3}d{e}^{3}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+12\,{\frac{{a}^{2}{d}^{2}{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-8\,{\frac{a{d}^{3}e}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{{d}^{4}}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(e*x+d)^(7/2)/b-2/5/b^2*(e*x+d)^(5/2)*a*e+2/5/b*(e*x+d)^(5/2)*d+2/3/b^3*(e*x
+d)^(3/2)*a^2*e^2-4/3/b^2*(e*x+d)^(3/2)*a*d*e+2/3/b*(e*x+d)^(3/2)*d^2-2/b^4*e^3*
a^3*(e*x+d)^(1/2)+6/b^3*a^2*d*e^2*(e*x+d)^(1/2)-6/b^2*a*d^2*e*(e*x+d)^(1/2)+2/b*
d^3*(e*x+d)^(1/2)+2/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*a^4*e^4-8/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(
1/2))*a^3*d*e^3+12/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*a^2*d^2*e^2-8/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*a*d^3*e+2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*d^4

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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)*(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 = 0.302475, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, b^{4}}, -\frac{2 \,{\left (105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*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*(15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^2*b*d*e^2 - 105*a^
3*e^3 + 3*(22*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^2*e - 112*a*b^2*d*e^2 +
35*a^2*b*e^3)*x)*sqrt(e*x + d))/b^4, -2/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^
2*b*d*e^2 - a^3*e^3)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)
/b)) - (15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^2*b*d*e^2 - 105*a
^3*e^3 + 3*(22*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^2*e - 112*a*b^2*d*e^2 +
35*a^2*b*e^3)*x)*sqrt(e*x + d))/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((b*x+a)*(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 = 0.297943, size = 356, normalized size = 2.58 \[ \frac{2 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{2} + 105 \, \sqrt{x e + d} b^{6} d^{3} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} e - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d e - 315 \, \sqrt{x e + d} a b^{5} d^{2} e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} e^{2} + 315 \, \sqrt{x e + d} a^{2} b^{4} d e^{2} - 105 \, \sqrt{x e + d} a^{3} b^{3} e^{3}\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(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)*arctan
(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(15*(x
*e + d)^(7/2)*b^6 + 21*(x*e + d)^(5/2)*b^6*d + 35*(x*e + d)^(3/2)*b^6*d^2 + 105*
sqrt(x*e + d)*b^6*d^3 - 21*(x*e + d)^(5/2)*a*b^5*e - 70*(x*e + d)^(3/2)*a*b^5*d*
e - 315*sqrt(x*e + d)*a*b^5*d^2*e + 35*(x*e + d)^(3/2)*a^2*b^4*e^2 + 315*sqrt(x*
e + d)*a^2*b^4*d*e^2 - 105*sqrt(x*e + d)*a^3*b^3*e^3)/b^7