3.743 \(\int \frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=310 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}} \]

[Out]

((c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8
*c*d*g^2*Sqrt[d + e*x]) - ((c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(4*g^2*Sqrt[d + e*x]) + ((f + g*x)^(3/2)*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) + ((c*d*f - a*e*g)^3*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])])/(8*c^(3/2)*d^(3/2)*g^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])

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Rubi [A]  time = 1.41012, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

((c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8
*c*d*g^2*Sqrt[d + e*x]) - ((c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(4*g^2*Sqrt[d + e*x]) + ((f + g*x)^(3/2)*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) + ((c*d*f - a*e*g)^3*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])])/(8*c^(3/2)*d^(3/2)*g^(5/2)*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 = 120.398, size = 291, normalized size = 0.94 \[ \frac{\left (f + g x\right )^{\frac{3}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 g \left (d + e x\right )^{\frac{3}{2}}} + \frac{\left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 g^{2} \sqrt{d + e x}} + \frac{\sqrt{f + g x} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 c d g^{2} \sqrt{d + e x}} - \frac{\left (a e g - c d f\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{8 c^{\frac{3}{2}} d^{\frac{3}{2}} g^{\frac{5}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)

[Out]

(f + g*x)**(3/2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*g*(d + e*x
)**(3/2)) + (f + g*x)**(3/2)*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))/(4*g**2*sqrt(d + e*x)) + sqrt(f + g*x)*(a*e*g - c*d*f)**2*sqrt(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*c*d*g**2*sqrt(d + e*x)) - (a*e*g - c*d*f
)**3*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)))/(8*c**(3/2)*d**(3/2)*g**(5/2)*sqrt(d + e*x)
*sqrt(a*e + c*d*x))

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Mathematica [A]  time = 0.463889, size = 210, normalized size = 0.68 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \sqrt{f+g x} \left (3 a^2 e^2 g^2+2 a c d e g (4 f+7 g x)+c^2 d^2 \left (-3 f^2+2 f g x+8 g^2 x^2\right )\right )}{3 c d g^2 (a e+c d x)}+\frac{(c d f-a e g)^3 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{c^{3/2} d^{3/2} g^{5/2} (a e+c d x)^{3/2}}\right )}{16 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((2*Sqrt[f + g*x]*(3*a^2*e^2*g^2 + 2*a*c*d*e*g*
(4*f + 7*g*x) + c^2*d^2*(-3*f^2 + 2*f*g*x + 8*g^2*x^2)))/(3*c*d*g^2*(a*e + c*d*x
)) + ((c*d*f - a*e*g)^3*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^(3/2)*d^(3/2)*g^(5/2)*(a*e + c*d*x)^(3/2)))
)/(16*(d + e*x)^(3/2))

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Maple [B]  time = 0.028, size = 602, normalized size = 1.9 \[ -{\frac{1}{48\,cd{g}^{2}}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{g}^{3}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{3}{e}^{3}-9\,{g}^{2}\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{2}{e}^{2}fcd+9\,g\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ) ae{f}^{2}{c}^{2}{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{3}{c}^{3}{d}^{3}-16\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}-28\,{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xaecd\sqrt{dgc}-4\,g\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xf{c}^{2}{d}^{2}\sqrt{dgc}-6\,{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{2}{e}^{2}\sqrt{dgc}-16\,acdefg\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{dgc}+6\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{f}^{2}{c}^{2}{d}^{2}\sqrt{dgc} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdg{x}^{2}+aegx+cdfx+aef}}}{\frac{1}{\sqrt{dgc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-1/48*(g*x+f)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*g^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^3*e^3-9*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*f*c*d+9*g*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*f^2*c^2*d^2-3*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))*f^3*c^3*d^3-16*x^2*c^2*d^2*g^2*(c*d
*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c)^(1/2)-28*g^2*(c*d*g*x^2+a*e*g*x+c*d*
f*x+a*e*f)^(1/2)*x*a*e*c*d*(d*g*c)^(1/2)-4*g*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(
1/2)*x*f*c^2*d^2*(d*g*c)^(1/2)-6*g^2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^2
*e^2*(d*g*c)^(1/2)-16*a*c*d*e*f*g*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(d*g*c
)^(1/2)+6*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*f^2*c^2*d^2*(d*g*c)^(1/2))/(e*
x+d)^(1/2)/(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)/c/d/(d*g*c)^(1/2)/g^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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04997, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(4*(8*c^2*d^2*g^2*x^2 - 3*c^2*d^2*f^2 + 8*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 2*
(c^2*d^2*f*g + 7*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(c^3*d^4*f^3 - 3*a*c^2*d^3*e*f^2*g + 3*
a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g + 3
*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*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*d
*e*g^2*x + c*d^2*g^2)*sqrt(c*d*g)), 1/48*(2*(8*c^2*d^2*g^2*x^2 - 3*c^2*d^2*f^2 +
8*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 2*(c^2*d^2*f*g + 7*a*c*d*e*g^2)*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) + 3*(c^
3*d^4*f^3 - 3*a*c^2*d^3*e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d
^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g + 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*x)*arctan(
2*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)/(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*d*e*g^2*x + c*d^2*g^2)*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)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}} \sqrt{g x + f}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)/(e*x + d)^
(3/2), x)