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

Optimal. Leaf size=133 \[ \frac{4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac{4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^4 d^2}-\frac{12 a \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^4 d^2} \]

[Out]

(-4*a*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^4*d^2) + (4*(3*a^2 - b^2*c
)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^4*d^2) - (12*a*(a + b*Sqrt[c + d*x])^(7/2))/
(7*b^4*d^2) + (4*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^4*d^2)

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Rubi [A]  time = 0.227356, antiderivative size = 133, 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 \[ \frac{4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac{4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^4 d^2}-\frac{12 a \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^4*d^2) + (4*(3*a^2 - b^2*c
)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^4*d^2) - (12*a*(a + b*Sqrt[c + d*x])^(7/2))/
(7*b^4*d^2) + (4*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^4*d^2)

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Rubi in Sympy [A]  time = 11.8205, size = 122, normalized size = 0.92 \[ - \frac{12 a \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}}}{7 b^{4} d^{2}} - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )}{3 b^{4} d^{2}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{9}{2}}}{9 b^{4} d^{2}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}} \left (3 a^{2} - b^{2} c\right )}{5 b^{4} d^{2}} \]

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))**(7/2)/(7*b**4*d**2) - 4*a*(a + b*sqrt(c + d*x))**(3
/2)*(a**2 - b**2*c)/(3*b**4*d**2) + 4*(a + b*sqrt(c + d*x))**(9/2)/(9*b**4*d**2)
+ 4*(a + b*sqrt(c + d*x))**(5/2)*(3*a**2 - b**2*c)/(5*b**4*d**2)

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Mathematica [A]  time = 0.0965162, size = 109, normalized size = 0.82 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-16 a^4+8 a^3 b \sqrt{c+d x}+6 a^2 b^2 (6 c-d x)+a b^3 \sqrt{c+d x} (5 d x-16 c)+7 b^4 \left (-4 c^2+c d x+5 d^2 x^2\right )\right )}{315 b^4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-16*a^4 + 6*a^2*b^2*(6*c - d*x) + 8*a^3*b*Sqrt[c +
d*x] + a*b^3*Sqrt[c + d*x]*(-16*c + 5*d*x) + 7*b^4*(-4*c^2 + c*d*x + 5*d^2*x^2)
))/(315*b^4*d^2)

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Maple [A]  time = 0.003, size = 94, normalized size = 0.7 \[ 4\,{\frac{1/9\, \left ( a+b\sqrt{dx+c} \right ) ^{9/2}-3/7\,a \left ( a+b\sqrt{dx+c} \right ) ^{7/2}+1/5\, \left ( -{b}^{2}c+3\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( -{b}^{2}c+{a}^{2} \right ) a \left ( a+b\sqrt{dx+c} \right ) ^{3/2}}{{b}^{4}{d}^{2}}} \]

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*(1/9*(a+b*(d*x+c)^(1/2))^(9/2)-3/7*a*(a+b*(d*x+c)^(1/2))^(7/2)+1/5*(-b
^2*c+3*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)-1/3*(-b^2*c+a^2)*a*(a+b*(d*x+c)^(1/2))^(3/
2))

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Maxima [A]  time = 0.71097, size = 126, normalized size = 0.95 \[ \frac{4 \,{\left (35 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} - 135 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} a - 63 \,{\left (b^{2} c - 3 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} + 105 \,{\left (a b^{2} c - a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}\right )}}{315 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/315*(35*(sqrt(d*x + c)*b + a)^(9/2) - 135*(sqrt(d*x + c)*b + a)^(7/2)*a - 63*(
b^2*c - 3*a^2)*(sqrt(d*x + c)*b + a)^(5/2) + 105*(a*b^2*c - a^3)*(sqrt(d*x + c)*
b + a)^(3/2))/(b^4*d^2)

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Fricas [A]  time = 0.352166, size = 139, normalized size = 1.05 \[ \frac{4 \,{\left (35 \, b^{4} d^{2} x^{2} - 28 \, b^{4} c^{2} + 36 \, a^{2} b^{2} c - 16 \, a^{4} +{\left (7 \, b^{4} c - 6 \, a^{2} b^{2}\right )} d x +{\left (5 \, a b^{3} d x - 16 \, a b^{3} c + 8 \, a^{3} b\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{315 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/315*(35*b^4*d^2*x^2 - 28*b^4*c^2 + 36*a^2*b^2*c - 16*a^4 + (7*b^4*c - 6*a^2*b^
2)*d*x + (5*a*b^3*d*x - 16*a*b^3*c + 8*a^3*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. \[ \int x \sqrt{a + b \sqrt{c + d x}}\, dx \]

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 = 0.302967, size = 460, normalized size = 3.46 \[ -\frac{4 \,{\left (63 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} b^{4} c{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 105 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a b^{4} c{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 35 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{4} b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 135 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{3} a b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 189 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} a^{2} b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + 105 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a^{3} b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}{\left | b \right |}}{315 \, b^{8} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/315*(63*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*b^4*c*sign((s
qrt(d*x + c)*b + a)*b - a*b) - 105*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c
)*b + a)*a*b^4*c*sign((sqrt(d*x + c)*b + a)*b - a*b) - 35*sqrt((sqrt(d*x + c)*b
+ a)*b^2)*(sqrt(d*x + c)*b + a)^4*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b) + 135*
sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a*b^2*sign((sqrt(d*x + c
)*b + a)*b - a*b) - 189*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*
a^2*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b) + 105*sqrt((sqrt(d*x + c)*b + a)*b^2
)*(sqrt(d*x + c)*b + a)*a^3*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b))*abs(b)/(b^8
*d^2)