3.2231 \(\int \frac{A+B x}{(a+b x)^{3/2} \sqrt{d+e x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.122113, 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 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} \sqrt{e}}-\frac{2 \sqrt{d+e x} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7586, size = 75, normalized size = 0.88 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{e}} + \frac{2 \sqrt{d + e x} \left (A b - B a\right )}{b \sqrt{a + b x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.165158, size = 97, normalized size = 1.14 \[ \frac{B \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{3/2} \sqrt{e}}-\frac{2 \sqrt{d+e x} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.034, size = 278, normalized size = 3.3 \[{\frac{1}{ \left ( ae-bd \right ) b}\sqrt{ex+d} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) xabe-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) x{b}^{2}d+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ){a}^{2}e-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) abd+2\,Ab\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-2\,Ba\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.512629, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (B a - A b\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B a b d - B a^{2} e +{\left (B b^{2} d - B a b e\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{2 \,{\left (a b^{2} d - a^{2} b e +{\left (b^{3} d - a b^{2} e\right )} x\right )} \sqrt{b e}}, \frac{2 \,{\left (B a - A b\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B a b d - B a^{2} e +{\left (B b^{2} d - B a b e\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{{\left (a b^{2} d - a^{2} b e +{\left (b^{3} d - a b^{2} e\right )} x\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.544814, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x