3.1712 \(\int \frac{(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=250 \[ -\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-35*e^3*Sqrt[d + e*x])/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*
x)^(3/2))/(96*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2
))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(7/2)/(4*b*(a
+ b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(9/2)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi [A]  time = 0.347307, antiderivative size = 250, 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 \[ -\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-35*e^3*Sqrt[d + e*x])/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*
x)^(3/2))/(96*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2
))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(7/2)/(4*b*(a
+ b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(9/2)*Sqrt[b*d - a*e]*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**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.430308, size = 159, normalized size = 0.64 \[ \frac{(a+b x)^5 \left (-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x} \left (326 e^2 (a+b x)^2 (b d-a e)+200 e (a+b x) (b d-a e)^2+48 (b d-a e)^3+279 e^3 (a+b x)^3\right )}{3 b^4 (a+b x)^4}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)^5*(-(Sqrt[d + e*x]*(48*(b*d - a*e)^3 + 200*e*(b*d - a*e)^2*(a + b*x)
+ 326*e^2*(b*d - a*e)*(a + b*x)^2 + 279*e^3*(a + b*x)^3))/(3*b^4*(a + b*x)^4) -
(35*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(9/2)*Sqrt[b*d - a*
e])))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.027, size = 467, normalized size = 1.9 \[ -{\frac{bx+a}{192\,{b}^{4}} \left ( -105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{4}{b}^{4}{e}^{4}-420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+279\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{7/2}{b}^{3}-630\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+511\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-511\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}d-420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+385\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-770\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+385\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}-105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}-315\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}+315\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e-105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/192*(-105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^4*e^4-420*arctan(
(e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^3*e^4+279*(b*(a*e-b*d))^(1/2)*(e*x+
d)^(7/2)*b^3-630*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*e^4+511
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-511*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)
*b^3*d-420*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b*e^4+385*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-770*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d
*e+385*(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+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)/(b*(a*e-b*d))^
(1/2)/b^4/((b*x+a)^2)^(5/2)

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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)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223215, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (279 \, b^{3} e^{3} x^{3} + 48 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e + 70 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} +{\left (326 \, b^{3} d e^{2} + 511 \, a b^{2} e^{3}\right )} x^{2} +{\left (200 \, b^{3} d^{2} e + 252 \, a b^{2} d e^{2} + 385 \, a^{2} b e^{3}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} - 105 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{384 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (279 \, b^{3} e^{3} x^{3} + 48 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e + 70 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} +{\left (326 \, b^{3} d e^{2} + 511 \, a b^{2} e^{3}\right )} x^{2} +{\left (200 \, b^{3} d^{2} e + 252 \, a b^{2} d e^{2} + 385 \, a^{2} b e^{3}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} + 105 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{192 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(2*(279*b^3*e^3*x^3 + 48*b^3*d^3 + 56*a*b^2*d^2*e + 70*a^2*b*d*e^2 + 105
*a^3*e^3 + (326*b^3*d*e^2 + 511*a*b^2*e^3)*x^2 + (200*b^3*d^2*e + 252*a*b^2*d*e^
2 + 385*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)))/((b^8*
x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)*sqrt(b^2*d - a*b*e)),
-1/192*((279*b^3*e^3*x^3 + 48*b^3*d^3 + 56*a*b^2*d^2*e + 70*a^2*b*d*e^2 + 105*a
^3*e^3 + (326*b^3*d*e^2 + 511*a*b^2*e^3)*x^2 + (200*b^3*d^2*e + 252*a*b^2*d*e^2
+ 385*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))))/((b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 +
4*a^3*b^5*x + a^4*b^4)*sqrt(-b^2*d + a*b*e))]

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24513, size = 390, normalized size = 1.56 \[ -\frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \, \sqrt{-b^{2} d + a b e} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{279 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 511 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 105 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 511 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 770 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 315 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 315 \, \sqrt{x e + d} a^{2} b d e^{6} + 105 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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)^(5/2),x, algorithm="giac")

[Out]

-35/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/(sqrt(-b^2*d + a*b*e)*b^
4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/192*(279*(x*e + d)^(7/2)*b^3*e^4 - 5
11*(x*e + d)^(5/2)*b^3*d*e^4 + 385*(x*e + d)^(3/2)*b^3*d^2*e^4 - 105*sqrt(x*e +
d)*b^3*d^3*e^4 + 511*(x*e + d)^(5/2)*a*b^2*e^5 - 770*(x*e + d)^(3/2)*a*b^2*d*e^5
+ 315*sqrt(x*e + d)*a*b^2*d^2*e^5 + 385*(x*e + d)^(3/2)*a^2*b*e^6 - 315*sqrt(x*
e + d)*a^2*b*d*e^6 + 105*sqrt(x*e + d)*a^3*e^7)/(((x*e + d)*b - b*d + a*e)^4*b^4
*sign(-(x*e + d)*b*e + b*d*e - a*e^2))