3.733 \(\int \frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=385 \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{24 \sqrt{d+e x}} \]

[Out]

(-5*(c*d*f - a*e*g)^3*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
/(64*c^3*d^3*g*Sqrt[d + e*x]) - (5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*c^2*d^2*g*Sqrt[d + e*x]) + (((a*e)/(c*d) -
f/g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*Sqrt[d +
e*x]) + ((f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*Sqrt[
d + e*x]) - (5*(c*d*f - a*e*g)^4*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g
]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(64*c^(7/2)*d^(7/2)*g^(3/
2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 1.9036, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104 \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{24 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(-5*(c*d*f - a*e*g)^3*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
/(64*c^3*d^3*g*Sqrt[d + e*x]) - (5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*c^2*d^2*g*Sqrt[d + e*x]) + (((a*e)/(c*d) -
f/g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*Sqrt[d +
e*x]) + ((f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*Sqrt[
d + e*x]) - (5*(c*d*f - a*e*g)^4*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g
]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(64*c^(7/2)*d^(7/2)*g^(3/
2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 157.767, size = 364, normalized size = 0.95 \[ - \frac{\left (f + g x\right )^{\frac{5}{2}} \left (- \frac{a e}{24 c d} + \frac{f}{24 g}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x}} + \frac{\left (f + g x\right )^{\frac{7}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 g \sqrt{d + e x}} - \frac{5 \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{96 c^{2} d^{2} g \sqrt{d + e x}} + \frac{5 \sqrt{f + g x} \left (a e g - c d f\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 c^{3} d^{3} g \sqrt{d + e x}} - \frac{5 \left (a e g - c d f\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{64 c^{\frac{7}{2}} d^{\frac{7}{2}} g^{\frac{3}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-(f + g*x)**(5/2)*(-a*e/(24*c*d) + f/(24*g))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))/sqrt(d + e*x) + (f + g*x)**(7/2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))/(4*g*sqrt(d + e*x)) - 5*(f + g*x)**(3/2)*(a*e*g - c*d*f)**2*sqrt(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(96*c**2*d**2*g*sqrt(d + e*x)) + 5*sqrt(f
+ g*x)*(a*e*g - c*d*f)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*c*
*3*d**3*g*sqrt(d + e*x)) - 5*(a*e*g - c*d*f)**4*sqrt(a*d*e + c*d*e*x**2 + x*(a*e
**2 + c*d**2))*atanh(sqrt(c)*sqrt(d)*sqrt(f + g*x)/(sqrt(g)*sqrt(a*e + c*d*x)))/
(64*c**(7/2)*d**(7/2)*g**(3/2)*sqrt(d + e*x)*sqrt(a*e + c*d*x))

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Mathematica [A]  time = 0.409775, size = 247, normalized size = 0.64 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{2 \sqrt{f+g x} \left (15 a^3 e^3 g^3-5 a^2 c d e^2 g^2 (11 f+2 g x)+a c^2 d^2 e g \left (73 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (15 f^3+118 f^2 g x+136 f g^2 x^2+48 g^3 x^3\right )\right )}{3 c^3 d^3 g}-\frac{5 (c d f-a e g)^4 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{c^{7/2} d^{7/2} g^{3/2} \sqrt{a e+c d x}}\right )}{128 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((2*Sqrt[f + g*x]*(15*a^3*e^3*g^3 - 5*a^2*c*d*e^2
*g^2*(11*f + 2*g*x) + a*c^2*d^2*e*g*(73*f^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(1
5*f^3 + 118*f^2*g*x + 136*f*g^2*x^2 + 48*g^3*x^3)))/(3*c^3*d^3*g) - (5*(c*d*f -
a*e*g)^4*Log[a*e*g + 2*Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqrt[f + g*x] +
c*d*(f + 2*g*x)])/(c^(7/2)*d^(7/2)*g^(3/2)*Sqrt[a*e + c*d*x])))/(128*Sqrt[d + e
*x])

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Maple [B]  time = 0.032, size = 870, normalized size = 2.3 \[ -{\frac{1}{384\,g{c}^{3}{d}^{3}}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( -96\,{x}^{3}{c}^{3}{d}^{3}{g}^{3}\sqrt{dgc}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}+15\,{g}^{4}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{4}{e}^{4}-60\,{g}^{3}{a}^{3}{e}^{3}f\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ) cd+90\,{f}^{2}{g}^{2}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{2}{e}^{2}{c}^{2}{d}^{2}-60\,{f}^{3}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ) aeg{c}^{3}{d}^{3}+15\,{f}^{4}{d}^{4}{c}^{4}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ) -16\,{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}\sqrt{dgc}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}-272\,{x}^{2}{c}^{3}{d}^{3}f{g}^{2}\sqrt{dgc}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}+20\,{g}^{3}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}x{a}^{2}{e}^{2}\sqrt{dgc}cd-72\,{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xaef\sqrt{dgc}{c}^{2}{d}^{2}-236\,{f}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xg\sqrt{dgc}{c}^{3}{d}^{3}-30\,{g}^{3}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{3}{e}^{3}\sqrt{dgc}+110\,{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{2}{e}^{2}f\sqrt{dgc}cd-146\,{f}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}aeg\sqrt{dgc}{c}^{2}{d}^{2}-30\,{f}^{3}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}{c}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdg{x}^{2}+aegx+cdfx+aef}}}{\frac{1}{\sqrt{dgc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/384*(g*x+f)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-96*x^3*c^3*d^3*g^
3*(d*g*c)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)+15*g^4*ln(1/2*(2*x*c*d*g
+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1
/2))*a^4*e^4-60*g^3*a^3*e^3*f*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x
+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*c*d+90*f^2*g^2*ln(1/2*(2*x*c
*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2))/(d*g*c
)^(1/2))*a^2*e^2*c^2*d^2-60*f^3*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g
*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a*e*g*c^3*d^3+15*f^4*d^4*c
^4*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*
c)^(1/2))/(d*g*c)^(1/2))-16*x^2*a*c^2*d^2*e*g^3*(d*g*c)^(1/2)*(c*d*g*x^2+a*e*g*x
+c*d*f*x+a*e*f)^(1/2)-272*x^2*c^3*d^3*f*g^2*(d*g*c)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d
*f*x+a*e*f)^(1/2)+20*g^3*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a^2*e^2*(d*g*
c)^(1/2)*c*d-72*g^2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a*e*f*(d*g*c)^(1/2
)*c^2*d^2-236*f^2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*g*(d*g*c)^(1/2)*c^3*
d^3-30*g^3*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^3*e^3*(d*g*c)^(1/2)+110*g^2
*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^2*e^2*f*(d*g*c)^(1/2)*c*d-146*f^2*(c*
d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a*e*g*(d*g*c)^(1/2)*c^2*d^2-30*f^3*(c*d*g*x
^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2)*c^3*d^3)/(e*x+d)^(1/2)/(c*d*g*x^2+
a*e*g*x+c*d*f*x+a*e*f)^(1/2)/g/(d*g*c)^(1/2)/c^3/d^3

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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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24216, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)/sqrt(e*x + d),x, algorithm="fricas")

[Out]

[1/768*(4*(48*c^3*d^3*g^3*x^3 + 15*c^3*d^3*f^3 + 73*a*c^2*d^2*e*f^2*g - 55*a^2*c
*d*e^2*f*g^2 + 15*a^3*e^3*g^3 + 8*(17*c^3*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 2*(
59*c^3*d^3*f^2*g + 18*a*c^2*d^2*e*f*g^2 - 5*a^2*c*d*e^2*g^3)*x)*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 15*(c^4*d^
5*f^4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3
+ a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 - 4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2
*g^2 - 4*a^3*c*d*e^4*f*g^3 + a^4*e^5*g^4)*x)*log((4*(2*c^2*d^2*g^2*x + c^2*d^2*f
*g + a*c*d*e*g^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt
(g*x + f) - (8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2
+ 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (c^2*d^2*e*f^2 + 2*(4*c^2
*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)*sqrt(c*d*g))/(e*x + d)
))/((c^3*d^3*e*g*x + c^3*d^4*g)*sqrt(c*d*g)), 1/384*(2*(48*c^3*d^3*g^3*x^3 + 15*
c^3*d^3*f^3 + 73*a*c^2*d^2*e*f^2*g - 55*a^2*c*d*e^2*f*g^2 + 15*a^3*e^3*g^3 + 8*(
17*c^3*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 2*(59*c^3*d^3*f^2*g + 18*a*c^2*d^2*e*f
*g^2 - 5*a^2*c*d*e^2*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c
*d*g)*sqrt(e*x + d)*sqrt(g*x + f) - 15*(c^4*d^5*f^4 - 4*a*c^3*d^4*e*f^3*g + 6*a^
2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3 + a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 -
4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a^4*e
^5*g^4)*x)*arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqr
t(e*x + d)*sqrt(g*x + f)/(2*c*d*e*g*x^2 + c*d^2*f + a*d*e*g + (c*d*e*f + (2*c*d^
2 + a*e^2)*g)*x)))/((c^3*d^3*e*g*x + c^3*d^4*g)*sqrt(-c*d*g))]

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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((g*x+f)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)/sqrt(e*x + d),x, algorithm="giac")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)/sqrt(e*x +
d), x)