3.2239 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{d+e x}}{b^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.166819, antiderivative size = 111, 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 \[ -\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{d+e x}}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5122, size = 100, normalized size = 0.9 \[ - \frac{2 B \sqrt{d + e x}}{b^{2} \sqrt{a + b x}} + \frac{2 B \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{b^{\frac{5}{2}}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.189448, size = 128, normalized size = 1.15 \[ \frac{B \sqrt{e} \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{5/2}}-\frac{2 \sqrt{d+e x} \left (B \left (-3 a^2 e+2 a b (d-2 e x)+3 b^2 d x\right )+A b^2 (d+e x)\right )}{3 b^2 (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 503, normalized size = 4.5 \[{\frac{1}{ \left ( 3\,ae-3\,bd \right ){b}^{2}} \left ( 3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}a{b}^{2}{e}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}{b}^{3}de+6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{a}^{2}b{e}^{2}-6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xa{b}^{2}de+2\,Ax{b}^{2}e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}{e}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}bde-8\,Bxabe\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+6\,Bx{b}^{2}d\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+2\,A{b}^{2}d\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-6\,B{a}^{2}e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+4\,Babd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{ex+d}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^(5/2),x)

[Out]

1/3*(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^2*a*b^2*e^2-3*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^2*b^3*d*e+6*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^2*b*e^2-6*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^2*d*e+2*A*x*b^2*e*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)+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^3*e^2-3*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*b*d*e-8*B*x*a*b*e*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+6*B*x*b^2*d*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+2*A*b^2*d*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)-6*B*a^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+4*B*a*b*d*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))*(e*x+d)^(1/2)/(b*e)^(1/2)/(a*e-b*d)/((b*x+a)
*(e*x+d))^(1/2)/b^2/(b*x+a)^(3/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.573645, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} b d - B a^{3} e +{\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \,{\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt{\frac{e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{e}{b}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (3 \, B a^{2} e -{\left (2 \, B a b + A b^{2}\right )} d -{\left (3 \, B b^{2} d -{\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}, \frac{3 \,{\left (B a^{2} b d - B a^{3} e +{\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \,{\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} b \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (3 \, B a^{2} e -{\left (2 \, B a b + A b^{2}\right )} d -{\left (3 \, B b^{2} d -{\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(B*a^2*b*d - B*a^3*e + (B*b^3*d - B*a*b^2*e)*x^2 + 2*(B*a*b^2*d - B*a^2*
b*e)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b^2*e
*x + b^2*d + a*b*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2
)*x) + 4*(3*B*a^2*e - (2*B*a*b + A*b^2)*d - (3*B*b^2*d - (4*B*a*b - A*b^2)*e)*x)
*sqrt(b*x + a)*sqrt(e*x + d))/(a^2*b^3*d - a^3*b^2*e + (b^5*d - a*b^4*e)*x^2 + 2
*(a*b^4*d - a^2*b^3*e)*x), 1/3*(3*(B*a^2*b*d - B*a^3*e + (B*b^3*d - B*a*b^2*e)*x
^2 + 2*(B*a*b^2*d - B*a^2*b*e)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)/(s
qrt(b*x + a)*sqrt(e*x + d)*b*sqrt(-e/b))) + 2*(3*B*a^2*e - (2*B*a*b + A*b^2)*d -
(3*B*b^2*d - (4*B*a*b - A*b^2)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(a^2*b^3*d -
a^3*b^2*e + (b^5*d - a*b^4*e)*x^2 + 2*(a*b^4*d - a^2*b^3*e)*x)]

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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)*(e*x+d)**(1/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.578963, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x