3.846 \(\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ \frac{8 (2 b d-a e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} \sqrt{e}}+\frac{6 d^2 \sqrt{a+b x}}{\sqrt{d+e x} (b d-a e)}+\frac{8 \sqrt{a+b x} \sqrt{d+e x}}{b} \]

[Out]

(6*d^2*Sqrt[a + b*x])/((b*d - a*e)*Sqrt[d + e*x]) + (8*Sqrt[a + b*x]*Sqrt[d + e*
x])/b + (8*(2*b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])
])/(b^(3/2)*Sqrt[e])

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Rubi [A]  time = 0.272959, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{8 (2 b d-a e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} \sqrt{e}}+\frac{6 d^2 \sqrt{a+b x}}{\sqrt{d+e x} (b d-a e)}+\frac{8 \sqrt{a+b x} \sqrt{d+e x}}{b} \]

Antiderivative was successfully verified.

[In]  Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(3/2)),x]

[Out]

(6*d^2*Sqrt[a + b*x])/((b*d - a*e)*Sqrt[d + e*x]) + (8*Sqrt[a + b*x]*Sqrt[d + e*
x])/b + (8*(2*b*d - a*e)*ArcTanh[(Sqrt[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 = 45.332, size = 139, normalized size = 1.29 \[ - \frac{6 d^{2} \sqrt{a + b x}}{\sqrt{d + e x} \left (a e - b d\right )} + \frac{8 \sqrt{a + b x} \sqrt{d + e x}}{b} + \frac{40 d \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{\sqrt{b} \sqrt{e}} - \frac{8 \left (a e + 3 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(3/2)/(b*x+a)**(1/2),x)

[Out]

-6*d**2*sqrt(a + b*x)/(sqrt(d + e*x)*(a*e - b*d)) + 8*sqrt(a + b*x)*sqrt(d + e*x
)/b + 40*d*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(sqrt(b)*sqrt(e)
) - 8*(a*e + 3*b*d)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(b**(3/
2)*sqrt(e))

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Mathematica [A]  time = 0.206155, size = 112, normalized size = 1.04 \[ \frac{4 (2 b d-a e) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{3/2} \sqrt{e}}+\sqrt{a+b x} \sqrt{d+e x} \left (\frac{8}{b}-\frac{6 d^2}{(d+e x) (a e-b d)}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(3/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[d + e*x]*(8/b - (6*d^2)/((-(b*d) + a*e)*(d + e*x))) + (4*(2*b
*d - a*e)*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 = 0.049, size = 438, normalized size = 4.1 \[ -2\,{\frac{\sqrt{bx+a}}{\sqrt{be}b \left ( ae-bd \right ) \sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{ex+d}} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{a}^{2}{e}^{3}-6\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xabd{e}^{2}+4\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{b}^{2}{d}^{2}e+2\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}d{e}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ab{d}^{2}e+4\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{3}-4\,xa{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+4\,xbde\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-4\,ade\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+7\,b{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(3/2)/(b*x+a)^(1/2),x)

[Out]

-2*(b*x+a)^(1/2)*(2*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*
d)/(b*e)^(1/2))*x*a^2*e^3-6*ln(1/2*(2*b*e*x+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+4*ln(1/2*(2*b*e*x+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*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*d*e^2-6*ln(1/2*(2*b*e*x+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+4*ln(1/2*(2*b*e*x+
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^3-4*x*a*e^2*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+4*x*b*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-
4*a*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+7*b*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2))/(b*e)^(1/2)/b/(a*e-b*d)/((b*x+a)*(e*x+d))^(1/2)/(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((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.391309, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left ({\left (7 \, b d^{2} - 4 \, a d e + 4 \,{\left (b d e - a e^{2}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} +{\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )\right )}}{{\left (b^{2} d^{2} - a b d e +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{b e}}, \frac{2 \,{\left ({\left (7 \, b d^{2} - 4 \, a d e + 4 \,{\left (b d e - a e^{2}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} + 2 \,{\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} +{\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )\right )}}{{\left (b^{2} d^{2} - a b d e +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[2*((7*b*d^2 - 4*a*d*e + 4*(b*d*e - a*e^2)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x +
d) - (2*b^2*d^3 - 3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*a*b*d*e^2 + a^2*e^
3)*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^2*d^2 - a*b*d*e + (b^2*d*e - a*b*e^2)*x)*sqrt(b*e)), 2*((7*b*d^2 - 4*a*d*
e + 4*(b*d*e - a*e^2)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 2*(2*b^2*d^3 -
3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*a*b*d*e^2 + a^2*e^3)*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^2*d^2 -
a*b*d*e + (b^2*d*e - a*b*e^2)*x)*sqrt(-b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(3/2)/(b*x+a)**(1/2),x)

[Out]

Integral((15*d**2 + 20*d*e*x + 8*e**2*x**2)/(sqrt(a + b*x)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.29906, size = 261, normalized size = 2.42 \[ -\frac{8 \,{\left (2 \, b d - a e\right )} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}{\left | b \right |}} + \frac{2 \, \sqrt{b x + a}{\left (\frac{4 \,{\left (b^{3} d e^{3} - a b^{2} e^{4}\right )}{\left (b x + a\right )}}{b^{3} d{\left | b \right |} e^{2} - a b^{2}{\left | b \right |} e^{3}} + \frac{7 \, b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 4 \, a^{2} b^{2} e^{4}}{b^{3} d{\left | b \right |} e^{2} - a b^{2}{\left | b \right |} e^{3}}\right )}}{\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

-8*(2*b*d - a*e)*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)))/(sqrt(b)*abs(b)) + 2*sqrt(b*x + a)*(4*(b^3*d*e^3 - a*b^2
*e^4)*(b*x + a)/(b^3*d*abs(b)*e^2 - a*b^2*abs(b)*e^3) + (7*b^4*d^2*e^2 - 8*a*b^3
*d*e^3 + 4*a^2*b^2*e^4)/(b^3*d*abs(b)*e^2 - a*b^2*abs(b)*e^3))/sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)