3.2223 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{3/2}}-\frac{2 \sqrt{a+b x} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.134716, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{3/2}}-\frac{2 \sqrt{a+b x} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.6122, size = 75, normalized size = 0.88 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{\sqrt{b} e^{\frac{3}{2}}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.157968, size = 97, normalized size = 1.14 \[ \frac{B \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{\sqrt{b} e^{3/2}}-\frac{2 \sqrt{a+b x} (A e-B d)}{e \sqrt{d+e x} (a e-b d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 278, normalized size = 3.3 \[{\frac{1}{e \left ( ae-bd \right ) } \left ( B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) xa{e}^{2}-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) xbde+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) ade-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) b{d}^{2}-2\,Ae\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+2\,Bd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.507909, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (B d - A e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (B b d^{2} - B a d e +{\left (B b d e - B a e^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{2 \,{\left (b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x\right )} \sqrt{b e}}, -\frac{2 \,{\left (B d - A e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (B b d^{2} - B a d e +{\left (B b d e - B a e^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{{\left (b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*(B*d - A*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) - (B*b*d^2 - B*a*d*e
+ (B*b*d*e - B*a*e^2)*x)*log(4*(2*b^2*e^2*x + b^2*d*e + a*b*e^2)*sqrt(b*x + a)*s
qrt(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*d^2*e - a*d*e^2 + (b*d*e^2 - a*e^3)*x)*sqrt(b*e)), -(2*
(B*d - A*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) - (B*b*d^2 - B*a*d*e + (B*b*d
*e - B*a*e^2)*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*d^2*e - a*d*e^2 + (b*d*e^2 - a*e^3)*x)*sqrt(-b*e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236359, size = 162, normalized size = 1.91 \[ -\frac{2 \, B{\left | b \right |} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{3}{2}}} - \frac{2 \,{\left (B b^{2} d{\left | b \right |} - A b^{2}{\left | b \right |} e\right )} \sqrt{b x + a}}{{\left (b^{3} d e - a b^{2} e^{2}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*B*abs(b)*e^(-3/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)))/b^(3/2) - 2*(B*b^2*d*abs(b) - A*b^2*abs(b)*e)*sqrt(b*x + a)/(
(b^3*d*e - a*b^2*e^2)*sqrt(b^2*d + (b*x + a)*b*e - a*b*e))