3.1657 \(\int \frac{(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=188 \[ \frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (a+b x) (b d-a e)}-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5} \]

[Out]

(-7*e^3*Sqrt[d + e*x])/(64*b^4*(a + b*x)^2) - (7*e^4*Sqrt[d + e*x])/(128*b^4*(b*
d - a*e)*(a + b*x)) - (7*e^2*(d + e*x)^(3/2))/(48*b^3*(a + b*x)^3) - (7*e*(d + e
*x)^(5/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(7/2)/(5*b*(a + b*x)^5) + (7*e^5*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(3/2))

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Rubi [A]  time = 0.322433, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (a+b x) (b d-a e)}-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-7*e^3*Sqrt[d + e*x])/(64*b^4*(a + b*x)^2) - (7*e^4*Sqrt[d + e*x])/(128*b^4*(b*
d - a*e)*(a + b*x)) - (7*e^2*(d + e*x)^(3/2))/(48*b^3*(a + b*x)^3) - (7*e*(d + e
*x)^(5/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(7/2)/(5*b*(a + b*x)^5) + (7*e^5*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(3/2))

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Rubi in Sympy [A]  time = 77.0821, size = 170, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{40 b^{2} \left (a + b x\right )^{4}} - \frac{7 e^{2} \left (d + e x\right )^{\frac{3}{2}}}{48 b^{3} \left (a + b x\right )^{3}} + \frac{7 e^{4} \sqrt{d + e x}}{128 b^{4} \left (a + b x\right ) \left (a e - b d\right )} - \frac{7 e^{3} \sqrt{d + e x}}{64 b^{4} \left (a + b x\right )^{2}} + \frac{7 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{9}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

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)**3,x)

[Out]

-(d + e*x)**(7/2)/(5*b*(a + b*x)**5) - 7*e*(d + e*x)**(5/2)/(40*b**2*(a + b*x)**
4) - 7*e**2*(d + e*x)**(3/2)/(48*b**3*(a + b*x)**3) + 7*e**4*sqrt(d + e*x)/(128*
b**4*(a + b*x)*(a*e - b*d)) - 7*e**3*sqrt(d + e*x)/(64*b**4*(a + b*x)**2) + 7*e*
*5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(9/2)*(a*e - b*d)**(3/2))

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Mathematica [A]  time = 0.386403, size = 171, normalized size = 0.91 \[ \frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (1210 e^3 (a+b x)^3 (b d-a e)+2104 e^2 (a+b x)^2 (b d-a e)^2+1488 e (a+b x) (b d-a e)^3+384 (b d-a e)^4+105 e^4 (a+b x)^4\right )}{1920 b^4 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(384*(b*d - a*e)^4 + 1488*e*(b*d - a*e)^3*(a + b*x) + 2104*e^2*(
b*d - a*e)^2*(a + b*x)^2 + 1210*e^3*(b*d - a*e)*(a + b*x)^3 + 105*e^4*(a + b*x)^
4))/(1920*b^4*(b*d - a*e)*(a + b*x)^5) + (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(3/2))

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Maple [B]  time = 0.028, size = 360, normalized size = 1.9 \[{\frac{7\,{e}^{5}}{128\, \left ( bex+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}-{\frac{79\,{e}^{5}}{192\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{e}^{6}a}{15\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{e}^{5}d}{15\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{49\,{a}^{2}{e}^{7}}{192\, \left ( bex+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{49\,{e}^{6}ad}{96\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{49\,{e}^{5}{d}^{2}}{192\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{8}{a}^{3}}{128\, \left ( bex+ae \right ) ^{5}{b}^{4}}\sqrt{ex+d}}+{\frac{21\,{a}^{2}{e}^{7}d}{128\, \left ( bex+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}-{\frac{21\,{e}^{6}a{d}^{2}}{128\, \left ( bex+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}+{\frac{7\,{e}^{5}{d}^{3}}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{7\,{e}^{5}}{ \left ( 128\,ae-128\,bd \right ){b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

7/128*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)-79/192*e^5/(b*e*x+a*e)^5/b*(e*x+
d)^(7/2)-7/15*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*a+7/15*e^5/(b*e*x+a*e)^5/b*(e*
x+d)^(5/2)*d-49/192*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*a^2+49/96*e^6/(b*e*x+a*e
)^5/b^2*(e*x+d)^(3/2)*a*d-49/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*d^2-7/128*e^8
/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*a^3+21/128*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*
a^2*d-21/128*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*a*d^2+7/128*e^5/(b*e*x+a*e)^5/b
*(e*x+d)^(1/2)*d^3+7/128*e^5/(a*e-b*d)/b^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((e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22951, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

[-1/3840*(2*(105*b^4*e^4*x^4 + 384*b^4*d^4 - 48*a*b^3*d^3*e - 56*a^2*b^2*d^2*e^2
- 70*a^3*b*d*e^3 - 105*a^4*e^4 + 10*(121*b^4*d*e^3 - 79*a*b^3*e^4)*x^3 + 2*(105
2*b^4*d^2*e^2 - 289*a*b^3*d*e^3 - 448*a^2*b^2*e^4)*x^2 + 2*(744*b^4*d^3*e - 128*
a*b^3*d^2*e^2 - 161*a^2*b^2*d*e^3 - 245*a^3*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e
*x + d) + 105*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e
^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*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)))/((a^5*b^5*d - a^6*b^4*e + (b^10*
d - a*b^9*e)*x^5 + 5*(a*b^9*d - a^2*b^8*e)*x^4 + 10*(a^2*b^8*d - a^3*b^7*e)*x^3
+ 10*(a^3*b^7*d - a^4*b^6*e)*x^2 + 5*(a^4*b^6*d - a^5*b^5*e)*x)*sqrt(b^2*d - a*b
*e)), -1/1920*((105*b^4*e^4*x^4 + 384*b^4*d^4 - 48*a*b^3*d^3*e - 56*a^2*b^2*d^2*
e^2 - 70*a^3*b*d*e^3 - 105*a^4*e^4 + 10*(121*b^4*d*e^3 - 79*a*b^3*e^4)*x^3 + 2*(
1052*b^4*d^2*e^2 - 289*a*b^3*d*e^3 - 448*a^2*b^2*e^4)*x^2 + 2*(744*b^4*d^3*e - 1
28*a*b^3*d^2*e^2 - 161*a^2*b^2*d*e^3 - 245*a^3*b*e^4)*x)*sqrt(-b^2*d + a*b*e)*sq
rt(e*x + d) - 105*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b
^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*
sqrt(e*x + d))))/((a^5*b^5*d - a^6*b^4*e + (b^10*d - a*b^9*e)*x^5 + 5*(a*b^9*d -
a^2*b^8*e)*x^4 + 10*(a^2*b^8*d - a^3*b^7*e)*x^3 + 10*(a^3*b^7*d - a^4*b^6*e)*x^
2 + 5*(a^4*b^6*d - a^5*b^5*e)*x)*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233111, size = 486, normalized size = 2.59 \[ -\frac{7 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d - a b^{4} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{105 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} + 790 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} - 896 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} + 490 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt{x e + d} b^{4} d^{4} e^{5} - 790 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} + 1792 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} - 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} - 896 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} + 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} - 490 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 420 \, \sqrt{x e + d} a^{3} b d e^{8} - 105 \, \sqrt{x e + d} a^{4} e^{9}}{1920 \,{\left (b^{5} d - a b^{4} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

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)^3,x, algorithm="giac")

[Out]

-7/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d - a*b^4*e)*sqrt(
-b^2*d + a*b*e)) - 1/1920*(105*(x*e + d)^(9/2)*b^4*e^5 + 790*(x*e + d)^(7/2)*b^4
*d*e^5 - 896*(x*e + d)^(5/2)*b^4*d^2*e^5 + 490*(x*e + d)^(3/2)*b^4*d^3*e^5 - 105
*sqrt(x*e + d)*b^4*d^4*e^5 - 790*(x*e + d)^(7/2)*a*b^3*e^6 + 1792*(x*e + d)^(5/2
)*a*b^3*d*e^6 - 1470*(x*e + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(x*e + d)*a*b^3*d^3
*e^6 - 896*(x*e + d)^(5/2)*a^2*b^2*e^7 + 1470*(x*e + d)^(3/2)*a^2*b^2*d*e^7 - 63
0*sqrt(x*e + d)*a^2*b^2*d^2*e^7 - 490*(x*e + d)^(3/2)*a^3*b*e^8 + 420*sqrt(x*e +
d)*a^3*b*d*e^8 - 105*sqrt(x*e + d)*a^4*e^9)/((b^5*d - a*b^4*e)*((x*e + d)*b - b
*d + a*e)^5)