3.1746 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=197 \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.404686, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.7661, size = 182, normalized size = 0.92 \[ - \frac{3 e \left (5 A b e - B a e - 4 B b d\right )}{4 b \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{5 A b e - B a e - 4 B b d}{4 b \left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \sqrt{d + e x} \left (a e - b d\right )} - \frac{3 e \left (5 A b e - B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \sqrt{b} \left (a e - b d\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.879222, size = 156, normalized size = 0.79 \[ \frac{1}{4} \left (\frac{\sqrt{d+e x} \left (\frac{2 (a B-A b) (b d-a e)}{(a+b x)^2}+\frac{-3 a B e+7 A b e-4 b B d}{a+b x}+\frac{8 e (A e-B d)}{d+e x}\right )}{(b d-a e)^3}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d + e*x]*((2*(-(A*b) + a*B)*(b*d - a*e))/(a + b*x)^2 + (-4*b*B*d + 7*A*b*
e - 3*a*B*e)/(a + b*x) + (8*e*(-(B*d) + A*e))/(d + e*x)))/(b*d - a*e)^3 + (3*e*(
4*b*B*d - 5*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sq
rt[b]*(b*d - a*e)^(7/2)))/4

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Maple [B]  time = 0.03, size = 485, normalized size = 2.5 \[ -2\,{\frac{A{e}^{2}}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+2\,{\frac{eBd}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,{b}^{2}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}Bde}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,Aab{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,{b}^{2}Ad{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,B{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{Bbad{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{{b}^{2}eB{d}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,Ab{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{3\,Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+3\,{\frac{bBde}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234017, 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*x + a)^3*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/8*(3*(4*B*a^2*b*d*e + (B*a^3 - 5*A*a^2*b)*e^2 + (4*B*b^3*d*e + (B*a*b^2 - 5*A
*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e + (B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d)*
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)) + 2*(8*A*a^2*e^2 - 2*(B*a*b + A*b^2)*d^2 - (13*B*a^2 - 9*A*a*b)*d*e
- 3*(4*B*b^2*d*e + (B*a*b - 5*A*b^2)*e^2)*x^2 - (4*B*b^2*d^2 + (21*B*a*b - 5*A*
b^2)*d*e + 5*(B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(b^2*d - a*b*e))/((a^2*b^3*d^3 - 3*a^
3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e
^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b
*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d)), 1/4*(3*(4*B*a^2*b*d*e + (B*a^3 - 5*
A*a^2*b)*e^2 + (4*B*b^3*d*e + (B*a*b^2 - 5*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e +
(B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a
*b*e)*sqrt(e*x + d))) + (8*A*a^2*e^2 - 2*(B*a*b + A*b^2)*d^2 - (13*B*a^2 - 9*A*a
*b)*d*e - 3*(4*B*b^2*d*e + (B*a*b - 5*A*b^2)*e^2)*x^2 - (4*B*b^2*d^2 + (21*B*a*b
- 5*A*b^2)*d*e + 5*(B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(-b^2*d + a*b*e))/((a^2*b^3*d^
3 - 3*a^3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2
*b^3*d*e^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2
- a^4*b*e^3)*x)*sqrt(-b^2*d + a*b*e)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224927, size = 467, normalized size = 2.37 \[ -\frac{3 \,{\left (4 \, B b d e + B a e^{2} - 5 \, A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (B d e - A e^{2}\right )}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{x e + d}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} - \sqrt{x e + d} B a b d e^{2} + 9 \, \sqrt{x e + d} A b^{2} d e^{2} + 5 \, \sqrt{x e + d} B a^{2} e^{3} - 9 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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