3.2230 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=145 \[ \frac{(-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (-3 a B e+2 A b e+b B d)}{b^2 (b d-a e)}-\frac{2 (d+e x)^{3/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.279715, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (-3 a B e+2 A b e+b B d)}{b^2 (b d-a e)}-\frac{2 (d+e x)^{3/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.0043, size = 136, normalized size = 0.94 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{b \sqrt{a + b x} \left (a e - b d\right )} - \frac{\sqrt{a + b x} \sqrt{d + e x} \left (2 A b e - 3 B a e + B b d\right )}{b^{2} \left (a e - b d\right )} + \frac{\left (2 A b e - 3 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(3/2),x)

[Out]

2*(d + e*x)**(3/2)*(A*b - B*a)/(b*sqrt(a + b*x)*(a*e - b*d)) - sqrt(a + b*x)*sqr
t(d + e*x)*(2*A*b*e - 3*B*a*e + B*b*d)/(b**2*(a*e - b*d)) + (2*A*b*e - 3*B*a*e +
B*b*d)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(b**(5/2)*sqrt(e))

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Mathematica [A]  time = 0.122778, size = 108, normalized size = 0.74 \[ \frac{(-3 a B e+2 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2 b^{5/2} \sqrt{e}}+\frac{\sqrt{d+e x} (3 a B-2 A b+b B x)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 386, normalized size = 2.7 \[{\frac{1}{2\,{b}^{2}}\sqrt{ex+d} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{b}^{2}e-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xabe+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) x{b}^{2}d+2\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) abe-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}e+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) abd+2\,Bxb\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-4\,Ab\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+6\,Ba\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^(3/2),x)

[Out]

1/2*(e*x+d)^(1/2)*(2*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e
+b*d)/(b*e)^(1/2))*x*b^2*e-3*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b*e+B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^2*d+2*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*e-3*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e+B*ln(1/2*(2*b*x*e+2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d+2*B*x*b*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)-4*A*b*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6*B*a*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/b^2/(b*x+a)^(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)*sqrt(e*x + d)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.584124, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (B b x + 3 \, B a - 2 \, A b\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B a b d -{\left (3 \, B a^{2} - 2 \, A a b\right )} e +{\left (B b^{2} d -{\left (3 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{4 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b e}}, \frac{2 \,{\left (B b x + 3 \, B a - 2 \, A b\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B a b d -{\left (3 \, B a^{2} - 2 \, A a b\right )} e +{\left (B b^{2} d -{\left (3 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{2 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*(B*b*x + 3*B*a - 2*A*b)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + (B*a*b*d
- (3*B*a^2 - 2*A*a*b)*e + (B*b^2*d - (3*B*a*b - 2*A*b^2)*e)*x)*log(4*(2*b^2*e^2
*x + b^2*d*e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d^2 +
6*a*b*d*e + a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/((b^3*x + a*b^2)*sqr
t(b*e)), 1/2*(2*(B*b*x + 3*B*a - 2*A*b)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) +
(B*a*b*d - (3*B*a^2 - 2*A*a*b)*e + (B*b^2*d - (3*B*a*b - 2*A*b^2)*e)*x)*arctan(
1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/((b^3*x
+ a*b^2)*sqrt(-b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{d + e x}}{\left (a + b x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(3/2),x)

[Out]

Integral((A + B*x)*sqrt(d + e*x)/(a + b*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.544686, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x