3.1740 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=221 \[ -\frac{b^{3/2} (5 a B e-7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}+\frac{b (5 a B e-7 A b e+2 b B d)}{\sqrt{d+e x} (b d-a e)^4}+\frac{5 a B e-7 A b e+2 b B d}{3 (d+e x)^{3/2} (b d-a e)^3}+\frac{5 a B e-7 A b e+2 b B d}{5 b (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{b (a+b x) (d+e x)^{5/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.52826, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^{3/2} (5 a B e-7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}+\frac{b (5 a B e-7 A b e+2 b B d)}{\sqrt{d+e x} (b d-a e)^4}+\frac{5 a B e-7 A b e+2 b B d}{3 (d+e x)^{3/2} (b d-a e)^3}+\frac{5 a B e-7 A b e+2 b B d}{5 b (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{b (a+b x) (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.8819, size = 207, normalized size = 0.94 \[ - \frac{b^{\frac{3}{2}} \left (7 A b e - 5 B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{9}{2}}} - \frac{b \left (7 A b e - 5 B a e - 2 B b d\right )}{\sqrt{d + e x} \left (a e - b d\right )^{4}} + \frac{7 A b e - 5 B a e - 2 B b d}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} - \frac{7 A b e - 5 B a e - 2 B b d}{5 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{b \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.758903, size = 191, normalized size = 0.86 \[ \frac{\sqrt{d+e x} \left (\frac{15 b^2 (a B-A b)}{a+b x}+\frac{30 b (2 a B e-3 A b e+b B d)}{d+e x}+\frac{10 (b d-a e) (a B e-2 A b e+b B d)}{(d+e x)^2}+\frac{6 (b d-a e)^2 (B d-A e)}{(d+e x)^3}\right )}{15 (b d-a e)^4}-\frac{b^{3/2} (5 a B e-7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((15*b^2*(-(A*b) + a*B))/(a + b*x) + (6*(b*d - a*e)^2*(B*d - A*e)
)/(d + e*x)^3 + (10*(b*d - a*e)*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x)^2 + (30*b*(
b*B*d - 3*A*b*e + 2*a*B*e))/(d + e*x)))/(15*(b*d - a*e)^4) - (b^(3/2)*(2*b*B*d -
7*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e
)^(9/2)

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Maple [B]  time = 0.033, size = 403, normalized size = 1.8 \[ -{\frac{2\,Ae}{5\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,Bd}{5\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,Abe}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,Bae}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,Bbd}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-6\,{\frac{A{b}^{2}e}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+4\,{\frac{Babe}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+2\,{\frac{{b}^{2}Bd}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-{\frac{A{b}^{3}e}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{Ba{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) }\sqrt{ex+d}}-7\,{\frac{A{b}^{3}e}{ \left ( ae-bd \right ) ^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+5\,{\frac{Ba{b}^{2}e}{ \left ( ae-bd \right ) ^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{B{b}^{3}d}{ \left ( ae-bd \right ) ^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*x + a)^2*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235421, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(12*A*a^3*e^3 - 2*(61*B*a*b^2 - 15*A*b^3)*d^3 - 8*(12*B*a^2*b - 29*A*a*b^
2)*d^2*e + 8*(B*a^3 - 8*A*a^2*b)*d*e^2 - 30*(2*B*b^3*d*e^2 + (5*B*a*b^2 - 7*A*b^
3)*e^3)*x^3 - 10*(14*B*b^3*d^2*e + (39*B*a*b^2 - 49*A*b^3)*d*e^2 + 2*(5*B*a^2*b
- 7*A*a*b^2)*e^3)*x^2 + 15*(2*B*a*b^2*d^3 + (5*B*a^2*b - 7*A*a*b^2)*d^2*e + (2*B
*b^3*d*e^2 + (5*B*a*b^2 - 7*A*b^3)*e^3)*x^3 + (4*B*b^3*d^2*e + 2*(6*B*a*b^2 - 7*
A*b^3)*d*e^2 + (5*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 + (2*B*b^3*d^3 + (9*B*a*b^2 - 7*
A*b^3)*d^2*e + 2*(5*B*a^2*b - 7*A*a*b^2)*d*e^2)*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)) - 2*(46*B*b^3*d^3 + (163*B*a*b^2 - 161*A*b^3)*d^2*e + 4*(29*B*a^2*b
- 42*A*a*b^2)*d*e^2 - 2*(5*B*a^3 - 7*A*a^2*b)*e^3)*x)/((a*b^4*d^6 - 4*a^2*b^3*d^
5*e + 6*a^3*b^2*d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4*e^2 - 4*a*b^4
*d^3*e^3 + 6*a^2*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e -
7*a*b^4*d^4*e^2 + 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e
^6)*x^2 + (b^5*d^6 - 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a
^4*b*d^2*e^4 + 2*a^5*d*e^5)*x)*sqrt(e*x + d)), -1/15*(6*A*a^3*e^3 - (61*B*a*b^2
- 15*A*b^3)*d^3 - 4*(12*B*a^2*b - 29*A*a*b^2)*d^2*e + 4*(B*a^3 - 8*A*a^2*b)*d*e^
2 - 15*(2*B*b^3*d*e^2 + (5*B*a*b^2 - 7*A*b^3)*e^3)*x^3 - 5*(14*B*b^3*d^2*e + (39
*B*a*b^2 - 49*A*b^3)*d*e^2 + 2*(5*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 + 15*(2*B*a*b^2*
d^3 + (5*B*a^2*b - 7*A*a*b^2)*d^2*e + (2*B*b^3*d*e^2 + (5*B*a*b^2 - 7*A*b^3)*e^3
)*x^3 + (4*B*b^3*d^2*e + 2*(6*B*a*b^2 - 7*A*b^3)*d*e^2 + (5*B*a^2*b - 7*A*a*b^2)
*e^3)*x^2 + (2*B*b^3*d^3 + (9*B*a*b^2 - 7*A*b^3)*d^2*e + 2*(5*B*a^2*b - 7*A*a*b^
2)*d*e^2)*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)) - (46*B*b^3*d^3 + (163*B*a*b^2 - 161*A*b^3)*d^2*e +
4*(29*B*a^2*b - 42*A*a*b^2)*d*e^2 - 2*(5*B*a^3 - 7*A*a^2*b)*e^3)*x)/((a*b^4*d^6
- 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4
*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 +
(2*b^5*d^5*e - 7*a*b^4*d^4*e^2 + 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b
*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 - 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2
*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5*d*e^5)*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)/(b*x+a)**2/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236931, size = 587, normalized size = 2.66 \[ \frac{{\left (2 \, B b^{3} d + 5 \, B a b^{2} e - 7 \, A b^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{\sqrt{x e + d} B a b^{2} e - \sqrt{x e + d} A b^{3} e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} B b^{2} d + 5 \,{\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} + 30 \,{\left (x e + d\right )}^{2} B a b e - 45 \,{\left (x e + d\right )}^{2} A b^{2} e - 10 \,{\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \,{\left (x e + d\right )} B a^{2} e^{2} + 10 \,{\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )}}{15 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*B*b^3*d + 5*B*a*b^2*e - 7*A*b^3*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e
))/((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)*sqrt
(-b^2*d + a*b*e)) + (sqrt(x*e + d)*B*a*b^2*e - sqrt(x*e + d)*A*b^3*e)/((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)*((x*e + d)*b - b*
d + a*e)) + 2/15*(15*(x*e + d)^2*B*b^2*d + 5*(x*e + d)*B*b^2*d^2 + 3*B*b^2*d^3 +
30*(x*e + d)^2*B*a*b*e - 45*(x*e + d)^2*A*b^2*e - 10*(x*e + d)*A*b^2*d*e - 6*B*
a*b*d^2*e - 3*A*b^2*d^2*e - 5*(x*e + d)*B*a^2*e^2 + 10*(x*e + d)*A*a*b*e^2 + 3*B
*a^2*d*e^2 + 6*A*a*b*d*e^2 - 3*A*a^2*e^3)/((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)*(x*e + d)^(5/2))