3.2188 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.169078, 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 \[ -\frac{2 (a+b x)^{3/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}+\frac{2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{5/2}}-\frac{2 B \sqrt{a+b x}}{e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3609, size = 100, normalized size = 0.9 \[ \frac{2 B \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{e^{\frac{5}{2}}} - \frac{2 B \sqrt{a + b x}}{e^{2} \sqrt{d + e x}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.181198, size = 128, normalized size = 1.15 \[ \frac{\sqrt{b} B \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{e^{5/2}}-\frac{2 \sqrt{a+b x} \left (a e (A e+2 B d+3 B e x)+A b e^2 x-b B d (3 d+4 e x)\right )}{3 e^2 (d+e x)^{3/2} (a e-b d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 503, normalized size = 4.5 \[ -{\frac{1}{ \left ( 3\,ae-3\,bd \right ){e}^{2}} \left ( -3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}ab{e}^{3}+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}{b}^{2}d{e}^{2}-6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xabd{e}^{2}+6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{b}^{2}{d}^{2}e+2\,Axb{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ab{d}^{2}e+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{3}+6\,Bxa{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-8\,Bxbde\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+2\,Aa{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+4\,Bade\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-6\,Bb{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(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*e^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*b^2*d*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*d*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*b^2*d^2*e+2*A*x*b*e^2*((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*b*d^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))*b^2*d^3+6*B*x*a*e^2*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)-8*B*x*b*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+2*A*a*e^2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+4*B*a*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-6*B*b*d^2
*((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^2/(e*x+d)^(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(b*x + a)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.560818, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B b d^{3} - B a d^{2} e +{\left (B b d e^{2} - B a e^{3}\right )} x^{2} + 2 \,{\left (B b d^{2} e - B a d e^{2}\right )} x\right )} \sqrt{\frac{b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{b}{e}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \,{\left (3 \, B b d^{2} - 2 \, B a d e - A a e^{2} +{\left (4 \, B b d e -{\left (3 \, B a + A b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (b d^{3} e^{2} - a d^{2} e^{3} +{\left (b d e^{4} - a e^{5}\right )} x^{2} + 2 \,{\left (b d^{2} e^{3} - a d e^{4}\right )} x\right )}}, \frac{3 \,{\left (B b d^{3} - B a d^{2} e +{\left (B b d e^{2} - B a e^{3}\right )} x^{2} + 2 \,{\left (B b d^{2} e - B a d e^{2}\right )} x\right )} \sqrt{-\frac{b}{e}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} e \sqrt{-\frac{b}{e}}}\right ) - 2 \,{\left (3 \, B b d^{2} - 2 \, B a d e - A a e^{2} +{\left (4 \, B b d e -{\left (3 \, B a + A b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (b d^{3} e^{2} - a d^{2} e^{3} +{\left (b d e^{4} - a e^{5}\right )} x^{2} + 2 \,{\left (b d^{2} e^{3} - a d e^{4}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.253881, size = 324, normalized size = 2.92 \[ \frac{B \sqrt{b}{\left | b \right |} e^{\frac{1}{2}}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{16 \,{\left (b^{6} d e^{4} - a b^{5} e^{5}\right )}} + \frac{\sqrt{b x + a}{\left (\frac{{\left (4 \, B b^{4} d{\left | b \right |} e^{2} - 3 \, B a b^{3}{\left | b \right |} e^{3} - A b^{4}{\left | b \right |} e^{3}\right )}{\left (b x + a\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}} + \frac{3 \,{\left (B b^{5} d^{2}{\left | b \right |} e - 2 \, B a b^{4} d{\left | b \right |} e^{2} + B a^{2} b^{3}{\left | b \right |} e^{3}\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}}\right )}}{48 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*B*sqrt(b)*abs(b)*e^(1/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e)))/(b^6*d*e^4 - a*b^5*e^5) + 1/48*sqrt(b*x + a)*((4*B*b
^4*d*abs(b)*e^2 - 3*B*a*b^3*abs(b)*e^3 - A*b^4*abs(b)*e^3)*(b*x + a)/(b^8*d^2*e^
4 - 2*a*b^7*d*e^5 + a^2*b^6*e^6) + 3*(B*b^5*d^2*abs(b)*e - 2*B*a*b^4*d*abs(b)*e^
2 + B*a^2*b^3*abs(b)*e^3)/(b^8*d^2*e^4 - 2*a*b^7*d*e^5 + a^2*b^6*e^6))/(b^2*d +
(b*x + a)*b*e - a*b*e)^(3/2)