3.2082 \(\int \frac{a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=196 \[ -\frac{63 b^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}}+\frac{63 b^2 e^2}{4 \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2}{4 (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2}{20 (d+e x)^{5/2} (b d-a e)^3}+\frac{9 e}{4 (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(63*e^2)/(20*(b*d - a*e)^3*(d + e*x)^(5/2)) - 1/(2*(b*d - a*e)*(a + b*x)^2*(d +
e*x)^(5/2)) + (9*e)/(4*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(5/2)) + (21*b*e^2)/(4*
(b*d - a*e)^4*(d + e*x)^(3/2)) + (63*b^2*e^2)/(4*(b*d - a*e)^5*Sqrt[d + e*x]) -
(63*b^(5/2)*e^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)
^(11/2))

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Rubi [A]  time = 0.328268, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{63 b^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}}+\frac{63 b^2 e^2}{4 \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2}{4 (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2}{20 (d+e x)^{5/2} (b d-a e)^3}+\frac{9 e}{4 (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(63*e^2)/(20*(b*d - a*e)^3*(d + e*x)^(5/2)) - 1/(2*(b*d - a*e)*(a + b*x)^2*(d +
e*x)^(5/2)) + (9*e)/(4*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(5/2)) + (21*b*e^2)/(4*
(b*d - a*e)^4*(d + e*x)^(3/2)) + (63*b^2*e^2)/(4*(b*d - a*e)^5*Sqrt[d + e*x]) -
(63*b^(5/2)*e^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)
^(11/2))

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Rubi in Sympy [A]  time = 94.8701, size = 175, normalized size = 0.89 \[ - \frac{63 b^{\frac{5}{2}} e^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{11}{2}}} - \frac{63 b^{2} e^{2}}{4 \sqrt{d + e x} \left (a e - b d\right )^{5}} + \frac{21 b e^{2}}{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} - \frac{63 e^{2}}{20 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{9 e}{4 \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

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)**2,x)

[Out]

-63*b**(5/2)*e**2*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(4*(a*e - b*d)**(1
1/2)) - 63*b**2*e**2/(4*sqrt(d + e*x)*(a*e - b*d)**5) + 21*b*e**2/(4*(d + e*x)**
(3/2)*(a*e - b*d)**4) - 63*e**2/(20*(d + e*x)**(5/2)*(a*e - b*d)**3) + 9*e/(4*(a
+ b*x)*(d + e*x)**(5/2)*(a*e - b*d)**2) + 1/(2*(a + b*x)**2*(d + e*x)**(5/2)*(a
*e - b*d))

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Mathematica [A]  time = 0.826211, size = 168, normalized size = 0.86 \[ \frac{\sqrt{d+e x} \left (-\frac{10 b^3 (b d-a e)}{(a+b x)^2}+\frac{75 b^3 e}{a+b x}+\frac{40 b e^2 (b d-a e)}{(d+e x)^2}+\frac{8 e^2 (b d-a e)^2}{(d+e x)^3}+\frac{240 b^2 e^2}{d+e x}\right )}{20 (b d-a e)^5}-\frac{63 b^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((-10*b^3*(b*d - a*e))/(a + b*x)^2 + (75*b^3*e)/(a + b*x) + (8*e^
2*(b*d - a*e)^2)/(d + e*x)^3 + (40*b*e^2*(b*d - a*e))/(d + e*x)^2 + (240*b^2*e^2
)/(d + e*x)))/(20*(b*d - a*e)^5) - (63*b^(5/2)*e^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(11/2))

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Maple [A]  time = 0.03, size = 231, normalized size = 1.2 \[ -{\frac{2\,{e}^{2}}{5\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-12\,{\frac{{b}^{2}{e}^{2}}{ \left ( ae-bd \right ) ^{5}\sqrt{ex+d}}}+2\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{3/2}}}-{\frac{15\,{e}^{2}{b}^{4}}{4\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{17\,{b}^{3}{e}^{3}a}{4\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{17\,{e}^{2}{b}^{4}d}{4\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{63\,{b}^{3}{e}^{2}}{4\, \left ( ae-bd \right ) ^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*e^2/(a*e-b*d)^3/(e*x+d)^(5/2)-12*e^2/(a*e-b*d)^5*b^2/(e*x+d)^(1/2)+2*e^2/(a
*e-b*d)^4*b/(e*x+d)^(3/2)-15/4*e^2/(a*e-b*d)^5*b^4/(b*e*x+a*e)^2*(e*x+d)^(3/2)-1
7/4*e^3/(a*e-b*d)^5*b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*a+17/4*e^2/(a*e-b*d)^5*b^4/(
b*e*x+a*e)^2*(e*x+d)^(1/2)*d-63/4*e^2/(a*e-b*d)^5*b^3/(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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.345465, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/40*(630*b^4*e^4*x^4 - 20*b^4*d^4 + 170*a*b^3*d^3*e + 576*a^2*b^2*d^2*e^2 - 11
2*a^3*b*d*e^3 + 16*a^4*e^4 + 210*(7*b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 42*(23*b^4*d^
2*e^2 + 59*a*b^3*d*e^3 + 8*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + a^2*b^2*d^2*e^2
+ 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x
^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log(
(b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a
)) + 6*(15*b^4*d^3*e + 277*a*b^3*d^2*e^2 + 136*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/(
(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6
*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 -
10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^
6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)
*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*
b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*
a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*
e^6)*x)*sqrt(e*x + d)), 1/20*(315*b^4*e^4*x^4 - 10*b^4*d^4 + 85*a*b^3*d^3*e + 28
8*a^2*b^2*d^2*e^2 - 56*a^3*b*d*e^3 + 8*a^4*e^4 + 105*(7*b^4*d*e^3 + 5*a*b^3*e^4)
*x^3 + 21*(23*b^4*d^2*e^2 + 59*a*b^3*d*e^3 + 8*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x
^4 + a^2*b^2*d^2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*
e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*sqrt(e*x + d)*sqrt
(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) + 3
*(15*b^4*d^3*e + 277*a*b^3*d^2*e^2 + 136*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^2*b
^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3
*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^
3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*
e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 +
(b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^
3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^
5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*x
)*sqrt(e*x + d))]

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293623, size = 512, normalized size = 2.61 \[ \frac{63 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{2} - 17 \, \sqrt{x e + d} b^{4} d e^{2} + 17 \, \sqrt{x e + d} a b^{3} e^{3}}{4 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} + \frac{2 \,{\left (30 \,{\left (x e + d\right )}^{2} b^{2} e^{2} + 5 \,{\left (x e + d\right )} b^{2} d e^{2} + b^{2} d^{2} e^{2} - 5 \,{\left (x e + d\right )} a b e^{3} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}}{5 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

63/4*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^5*d^5 - 5*a*b^4*d^
4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b
^2*d + a*b*e)) + 1/4*(15*(x*e + d)^(3/2)*b^4*e^2 - 17*sqrt(x*e + d)*b^4*d*e^2 +
17*sqrt(x*e + d)*a*b^3*e^3)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*
a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*((x*e + d)*b - b*d + a*e)^2) + 2/5*(3
0*(x*e + d)^2*b^2*e^2 + 5*(x*e + d)*b^2*d*e^2 + b^2*d^2*e^2 - 5*(x*e + d)*a*b*e^
3 - 2*a*b*d*e^3 + a^2*e^4)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a
^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(x*e + d)^(5/2))