3.1747 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=240 \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

[Out]

(-5*e*(4*b*B*d - 7*A*b*e + 3*a*B*e))/(12*b*(b*d - a*e)^3*(d + e*x)^(3/2)) - (A*b
- a*B)/(2*b*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 7*A*b*e + 3*a
*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)) - (5*e*(4*b*B*d - 7*A*b*e +
3*a*B*e))/(4*(b*d - a*e)^4*Sqrt[d + e*x]) + (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*
a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.51879, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]

[Out]

(-5*e*(4*b*B*d - 7*A*b*e + 3*a*B*e))/(12*b*(b*d - a*e)^3*(d + e*x)^(3/2)) - (A*b
- a*B)/(2*b*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 7*A*b*e + 3*a
*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)) - (5*e*(4*b*B*d - 7*A*b*e +
3*a*B*e))/(4*(b*d - a*e)^4*Sqrt[d + e*x]) + (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*
a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

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Rubi in Sympy [A]  time = 54.8479, size = 230, normalized size = 0.96 \[ \frac{5 \sqrt{b} e \left (7 A b e - 3 B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{9}{2}}} + \frac{5 e \left (7 A b e - 3 B a e - 4 B b d\right )}{4 \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{5 e \left (7 A b e - 3 B a e - 4 B b d\right )}{12 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{7 A b e - 3 B a e - 4 B b d}{4 b \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d)**(5/2),x)

[Out]

5*sqrt(b)*e*(7*A*b*e - 3*B*a*e - 4*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e -
b*d))/(4*(a*e - b*d)**(9/2)) + 5*e*(7*A*b*e - 3*B*a*e - 4*B*b*d)/(4*sqrt(d + e*x
)*(a*e - b*d)**4) - 5*e*(7*A*b*e - 3*B*a*e - 4*B*b*d)/(12*b*(d + e*x)**(3/2)*(a*
e - b*d)**3) + (7*A*b*e - 3*B*a*e - 4*B*b*d)/(4*b*(a + b*x)*(d + e*x)**(3/2)*(a*
e - b*d)**2) + (A*b - B*a)/(2*b*(a + b*x)**2*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 1.36517, size = 195, normalized size = 0.81 \[ \frac{\sqrt{d+e x} \left (-\frac{6 b (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{8 e (b d-a e) (A e-B d)}{(d+e x)^2}-\frac{3 b (7 a B e-11 A b e+4 b B d)}{a+b x}+\frac{24 e (-a B e+3 A b e-2 b B d)}{d+e x}\right )}{12 (b d-a e)^4}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]

[Out]

(Sqrt[d + e*x]*((-6*b*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2 - (3*b*(4*b*B*d - 11*
A*b*e + 7*a*B*e))/(a + b*x) + (8*e*(b*d - a*e)*(-(B*d) + A*e))/(d + e*x)^2 + (24
*e*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)))/(12*(b*d - a*e)^4) + (5*Sqrt[b]*e*(
4*b*B*d - 7*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
4*(b*d - a*e)^(9/2))

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Maple [B]  time = 0.037, size = 568, normalized size = 2.4 \[ -{\frac{2\,A{e}^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,eBd}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{Ab{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-2\,{\frac{Ba{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-4\,{\frac{bBde}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{b}^{3}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{3}Bde}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{b}^{2}Aa{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{b}^{3}Ad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{9\,bB{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,{b}^{2}Bad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{{b}^{3}B{d}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{b}^{2}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{15\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-5\,{\frac{{b}^{2}Bde}{ \left ( ae-bd \right ) ^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^3/(e*x+d)^(5/2),x)

[Out]

-2/3/(a*e-b*d)^3/(e*x+d)^(3/2)*A*e^2+2/3*e/(a*e-b*d)^3/(e*x+d)^(3/2)*B*d+6/(a*e-
b*d)^4/(e*x+d)^(1/2)*A*b*e^2-2/(a*e-b*d)^4/(e*x+d)^(1/2)*B*a*e^2-4*e/(a*e-b*d)^4
/(e*x+d)^(1/2)*B*b*d+11/4/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*e^2-7/4/
(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*e^2-e/(a*e-b*d)^4*b^3/(b*e*x+a*e
)^2*(e*x+d)^(3/2)*B*d+13/4/(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a*e^3-1
3/4/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*d*e^2-9/4/(a*e-b*d)^4*b/(b*e*x
+a*e)^2*(e*x+d)^(1/2)*B*a^2*e^3+5/4/(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*
B*a*d*e^2+e/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*d^2+35/4/(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))*A*e^2-15/4/(a
*e-b*d)^4*b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*
e^2-5*e/(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*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)^3*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240887, 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)^3*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(16*A*a^3*e^3 + 12*(B*a*b^2 + A*b^3)*d^3 + 2*(83*B*a^2*b - 39*A*a*b^2)*d^
2*e + 32*(B*a^3 - 5*A*a^2*b)*d*e^2 + 30*(4*B*b^3*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e
^3)*x^3 + 10*(16*B*b^3*d^2*e + 4*(8*B*a*b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*
A*a*b^2)*e^3)*x^2 + 15*(4*B*a^2*b*d^2*e + (3*B*a^3 - 7*A*a^2*b)*d*e^2 + (4*B*b^3
*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e^3)*x^3 + (4*B*b^3*d^2*e + (11*B*a*b^2 - 7*A*b^3
)*d*e^2 + 2*(3*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 + (8*B*a*b^2*d^2*e + 2*(5*B*a^2*b -
7*A*a*b^2)*d*e^2 + (3*B*a^3 - 7*A*a^2*b)*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)) + 2*(12*B*b^3*d^3 + (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(67*B*a^2*b -
119*A*a*b^2)*d*e^2 + 8*(3*B*a^3 - 7*A*a^2*b)*e^3)*x)/((a^2*b^4*d^5 - 4*a^3*b^3*d
^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^6*d^4*e - 4*a*b^5*d^
3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^6*d^5 - 2*a*
b^5*d^4*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^
5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3
- 2*a^5*b*d*e^4 + a^6*e^5)*x)*sqrt(e*x + d)), -1/12*(8*A*a^3*e^3 + 6*(B*a*b^2 +
A*b^3)*d^3 + (83*B*a^2*b - 39*A*a*b^2)*d^2*e + 16*(B*a^3 - 5*A*a^2*b)*d*e^2 + 15
*(4*B*b^3*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e^3)*x^3 + 5*(16*B*b^3*d^2*e + 4*(8*B*a*
b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 - 15*(4*B*a^2*b*d^2*e
+ (3*B*a^3 - 7*A*a^2*b)*d*e^2 + (4*B*b^3*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e^3)*x^3
+ (4*B*b^3*d^2*e + (11*B*a*b^2 - 7*A*b^3)*d*e^2 + 2*(3*B*a^2*b - 7*A*a*b^2)*e^3)
*x^2 + (8*B*a*b^2*d^2*e + 2*(5*B*a^2*b - 7*A*a*b^2)*d*e^2 + (3*B*a^3 - 7*A*a^2*b
)*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)) + (12*B*b^3*d^3 + (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(6
7*B*a^2*b - 119*A*a*b^2)*d*e^2 + 8*(3*B*a^3 - 7*A*a^2*b)*e^3)*x)/((a^2*b^4*d^5 -
4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^6*d^4*e
- 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^
6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4
+ 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*
b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*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)**3/(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227213, size = 606, normalized size = 2.52 \[ -\frac{5 \,{\left (4 \, B b^{2} d e + 3 \, B a b e^{2} - 7 \, A b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d e + B b d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - 9 \,{\left (x e + d\right )} A b e^{2} - B a d e^{2} - A b d e^{2} + A a e^{3}\right )}}{3 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e - 4 \, \sqrt{x e + d} B b^{3} d^{2} e + 7 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{2} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{2} - 5 \, \sqrt{x e + d} B a b^{2} d e^{2} + 13 \, \sqrt{x e + d} A b^{3} d e^{2} + 9 \, \sqrt{x e + d} B a^{2} b e^{3} - 13 \, \sqrt{x e + d} A a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-5/4*(4*B*b^2*d*e + 3*B*a*b*e^2 - 7*A*b^2*e^2)*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)) - 2/3*(6*(x*e + d)*B*b*d*e + B*b*d^2*e + 3*(x*e + d)*
B*a*e^2 - 9*(x*e + d)*A*b*e^2 - B*a*d*e^2 - A*b*d*e^2 + A*a*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)^(3/2)) - 1/4
*(4*(x*e + d)^(3/2)*B*b^3*d*e - 4*sqrt(x*e + d)*B*b^3*d^2*e + 7*(x*e + d)^(3/2)*
B*a*b^2*e^2 - 11*(x*e + d)^(3/2)*A*b^3*e^2 - 5*sqrt(x*e + d)*B*a*b^2*d*e^2 + 13*
sqrt(x*e + d)*A*b^3*d*e^2 + 9*sqrt(x*e + d)*B*a^2*b*e^3 - 13*sqrt(x*e + d)*A*a*b
^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)*b - b*d + a*e)^2)