3.487 \(\int \frac{x}{\sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=131 \[ \frac{4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^4 d^2}-\frac{4 a \left (a^2-b^2 c\right ) \sqrt{a+b \sqrt{c+d x}}}{b^4 d^2}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^4 d^2}-\frac{12 a \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^4 d^2} \]

[Out]

(-4*a*(a^2 - b^2*c)*Sqrt[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (4*(3*a^2 - b^2*c)*(a
+ b*Sqrt[c + d*x])^(3/2))/(3*b^4*d^2) - (12*a*(a + b*Sqrt[c + d*x])^(5/2))/(5*b
^4*d^2) + (4*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^4*d^2)

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Rubi [A]  time = 0.227495, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^4 d^2}-\frac{4 a \left (a^2-b^2 c\right ) \sqrt{a+b \sqrt{c+d x}}}{b^4 d^2}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^4 d^2}-\frac{12 a \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^4 d^2} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)*Sqrt[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (4*(3*a^2 - b^2*c)*(a
+ b*Sqrt[c + d*x])^(3/2))/(3*b^4*d^2) - (12*a*(a + b*Sqrt[c + d*x])^(5/2))/(5*b
^4*d^2) + (4*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^4*d^2)

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Rubi in Sympy [A]  time = 12.0638, size = 121, normalized size = 0.92 \[ - \frac{12 a \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}}}{5 b^{4} d^{2}} - \frac{4 a \sqrt{a + b \sqrt{c + d x}} \left (a^{2} - b^{2} c\right )}{b^{4} d^{2}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}}}{7 b^{4} d^{2}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (3 a^{2} - b^{2} c\right )}{3 b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*(d*x+c)**(1/2))**(1/2),x)

[Out]

-12*a*(a + b*sqrt(c + d*x))**(5/2)/(5*b**4*d**2) - 4*a*sqrt(a + b*sqrt(c + d*x))
*(a**2 - b**2*c)/(b**4*d**2) + 4*(a + b*sqrt(c + d*x))**(7/2)/(7*b**4*d**2) + 4*
(a + b*sqrt(c + d*x))**(3/2)*(3*a**2 - b**2*c)/(3*b**4*d**2)

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Mathematica [A]  time = 0.0896397, size = 84, normalized size = 0.64 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-48 a^3+24 a^2 b \sqrt{c+d x}+2 a b^2 (26 c-9 d x)+5 b^3 \sqrt{c+d x} (3 d x-4 c)\right )}{105 b^4 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-48*a^3 + 2*a*b^2*(26*c - 9*d*x) + 24*a^2*b*Sqrt[c
+ d*x] + 5*b^3*Sqrt[c + d*x]*(-4*c + 3*d*x)))/(105*b^4*d^2)

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Maple [A]  time = 0.003, size = 94, normalized size = 0.7 \[ 4\,{\frac{1/7\, \left ( a+b\sqrt{dx+c} \right ) ^{7/2}-3/5\, \left ( a+b\sqrt{dx+c} \right ) ^{5/2}a+1/3\, \left ( -{b}^{2}c+3\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{3/2}- \left ( -{b}^{2}c+{a}^{2} \right ) a\sqrt{a+b\sqrt{dx+c}}}{{b}^{4}{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(d*x+c)^(1/2))^(1/2),x)

[Out]

4/d^2/b^4*(1/7*(a+b*(d*x+c)^(1/2))^(7/2)-3/5*(a+b*(d*x+c)^(1/2))^(5/2)*a+1/3*(-b
^2*c+3*a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-(-b^2*c+a^2)*a*(a+b*(d*x+c)^(1/2))^(1/2))

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Maxima [A]  time = 0.699476, size = 126, normalized size = 0.96 \[ \frac{4 \,{\left (15 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} - 63 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} a - 35 \,{\left (b^{2} c - 3 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} + 105 \,{\left (a b^{2} c - a^{3}\right )} \sqrt{\sqrt{d x + c} b + a}\right )}}{105 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(sqrt(d*x + c)*b + a),x, algorithm="maxima")

[Out]

4/105*(15*(sqrt(d*x + c)*b + a)^(7/2) - 63*(sqrt(d*x + c)*b + a)^(5/2)*a - 35*(b
^2*c - 3*a^2)*(sqrt(d*x + c)*b + a)^(3/2) + 105*(a*b^2*c - a^3)*sqrt(sqrt(d*x +
c)*b + a))/(b^4*d^2)

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Fricas [A]  time = 0.330886, size = 96, normalized size = 0.73 \[ -\frac{4 \,{\left (18 \, a b^{2} d x - 52 \, a b^{2} c + 48 \, a^{3} -{\left (15 \, b^{3} d x - 20 \, b^{3} c + 24 \, a^{2} b\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{105 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(sqrt(d*x + c)*b + a),x, algorithm="fricas")

[Out]

-4/105*(18*a*b^2*d*x - 52*a*b^2*c + 48*a^3 - (15*b^3*d*x - 20*b^3*c + 24*a^2*b)*
sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^4*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + b \sqrt{c + d x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*(d*x+c)**(1/2))**(1/2),x)

[Out]

Integral(x/sqrt(a + b*sqrt(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.292773, size = 412, normalized size = 3.15 \[ -\frac{4 \,{\left (35 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} b^{2} c{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 105 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}} a b^{2} c{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 15 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{3}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 63 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} a{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 105 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 105 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}} a^{3}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}}{105 \, b^{4} d^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(sqrt(d*x + c)*b + a),x, algorithm="giac")

[Out]

-4/105*(35*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*b^2*c*sign((sqr
t(d*x + c)*b + a)*b - a*b) - 105*sqrt((sqrt(d*x + c)*b + a)*b^2)*a*b^2*c*sign((s
qrt(d*x + c)*b + a)*b - a*b) - 15*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)
*b + a)^3*sign((sqrt(d*x + c)*b + a)*b - a*b) + 63*sqrt((sqrt(d*x + c)*b + a)*b^
2)*(sqrt(d*x + c)*b + a)^2*a*sign((sqrt(d*x + c)*b + a)*b - a*b) - 105*sqrt((sqr
t(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^2*sign((sqrt(d*x + c)*b + a)*b -
a*b) + 105*sqrt((sqrt(d*x + c)*b + a)*b^2)*a^3*sign((sqrt(d*x + c)*b + a)*b - a*
b))/(b^4*d^2*abs(b))