3.1692 \(\int \frac{(d+e x)^{7/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=263 \[ \frac{2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.47282, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.313274, size = 166, normalized size = 0.63 \[ \frac{(a+b x) \left (\frac{2 \sqrt{d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 b^4}-\frac{2 (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((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)))/Sqrt[(a + b*x)^2]

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Maple [B]  time = 0.013, size = 462, normalized size = 1.8 \[{\frac{2\,bx+2\,a}{105\,{b}^{4}} \left ( 15\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{7/2}{b}^{3}-21\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e+21\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}d+35\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-70\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+35\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}-420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}bd{e}^{3}+630\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{3}{d}^{3}e+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{4}{d}^{4}-105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}+315\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}-315\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(b*x+a)*(15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^3-21*(b*(a*e-b*d))^(1/2)*(
e*x+d)^(5/2)*a*b^2*e+21*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^3*d+35*(b*(a*e-b*d))
^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-70*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+35
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*
d))^(1/2))*a^4*e^4-420*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3+6
30*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d^2*e^2-420*arctan((e*x+d
)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*b^3*d^3*e+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*
d))^(1/2))*b^4*d^4-105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+315*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2-315*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d
^2*e+105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)/((b*x+a)^2)^(1/2)/b^4/(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((e*x + d)^(7/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218691, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, b^{4}}, -\frac{2 \,{\left (105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \,{\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227032, size = 478, normalized size = 1.82 \[ \frac{2 \,{\left (b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{2}{\rm sign}\left (b x + a\right ) + 105 \, \sqrt{x e + d} b^{6} d^{3}{\rm sign}\left (b x + a\right ) - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} e{\rm sign}\left (b x + a\right ) - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d e{\rm sign}\left (b x + a\right ) - 315 \, \sqrt{x e + d} a b^{5} d^{2} e{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} e^{2}{\rm sign}\left (b x + a\right ) + 315 \, \sqrt{x e + d} a^{2} b^{4} d e^{2}{\rm sign}\left (b x + a\right ) - 105 \, \sqrt{x e + d} a^{3} b^{3} e^{3}{\rm sign}\left (b x + a\right )\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^4*d^4*sign(b*x + a) - 4*a*b^3*d^3*e*sign(b*x + a) + 6*a^2*b^2*d^2*e^2*sign(
b*x + a) - 4*a^3*b*d*e^3*sign(b*x + a) + a^4*e^4*sign(b*x + a))*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*sign(b*x + a) + 21*(x*e + d)^(5/2)*b^6*d*sign(b*x + a) + 35*(x*e + d)^(3
/2)*b^6*d^2*sign(b*x + a) + 105*sqrt(x*e + d)*b^6*d^3*sign(b*x + a) - 21*(x*e +
d)^(5/2)*a*b^5*e*sign(b*x + a) - 70*(x*e + d)^(3/2)*a*b^5*d*e*sign(b*x + a) - 31
5*sqrt(x*e + d)*a*b^5*d^2*e*sign(b*x + a) + 35*(x*e + d)^(3/2)*a^2*b^4*e^2*sign(
b*x + a) + 315*sqrt(x*e + d)*a^2*b^4*d*e^2*sign(b*x + a) - 105*sqrt(x*e + d)*a^3
*b^3*e^3*sign(b*x + a))/b^7