3.485 \(\int \frac{x^3}{\sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=324 \[ \frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4} \]

[Out]

(-4*a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]])/(b^8*d^4) + (4*(a^2 - b^2*c)^2*
(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*
c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) + (4*(35*a^4 - 30*a^2*
b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8*d^4) - (20*a*(7*a^2 - 3*b
^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt
[c + d*x])^(11/2))/(11*b^8*d^4) - (28*a*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^
4) + (4*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4)

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Rubi [A]  time = 0.532948, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]])/(b^8*d^4) + (4*(a^2 - b^2*c)^2*
(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*
c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) + (4*(35*a^4 - 30*a^2*
b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8*d^4) - (20*a*(7*a^2 - 3*b
^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt
[c + d*x])^(11/2))/(11*b^8*d^4) - (28*a*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^
4) + (4*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4)

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Rubi in Sympy [A]  time = 32.0165, size = 304, normalized size = 0.94 \[ - \frac{28 a \left (a + b \sqrt{c + d x}\right )^{\frac{13}{2}}}{13 b^{8} d^{4}} - \frac{20 a \left (a + b \sqrt{c + d x}\right )^{\frac{9}{2}} \left (7 a^{2} - 3 b^{2} c\right )}{9 b^{8} d^{4}} - \frac{12 a \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}} \left (a^{2} - b^{2} c\right ) \left (7 a^{2} - 3 b^{2} c\right )}{5 b^{8} d^{4}} - \frac{4 a \sqrt{a + b \sqrt{c + d x}} \left (a^{2} - b^{2} c\right )^{3}}{b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{15}{2}}}{15 b^{8} d^{4}} + \frac{12 \left (a + b \sqrt{c + d x}\right )^{\frac{11}{2}} \left (7 a^{2} - b^{2} c\right )}{11 b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}} \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{7 b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{2} \left (7 a^{2} - b^{2} c\right )}{3 b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-28*a*(a + b*sqrt(c + d*x))**(13/2)/(13*b**8*d**4) - 20*a*(a + b*sqrt(c + d*x))*
*(9/2)*(7*a**2 - 3*b**2*c)/(9*b**8*d**4) - 12*a*(a + b*sqrt(c + d*x))**(5/2)*(a*
*2 - b**2*c)*(7*a**2 - 3*b**2*c)/(5*b**8*d**4) - 4*a*sqrt(a + b*sqrt(c + d*x))*(
a**2 - b**2*c)**3/(b**8*d**4) + 4*(a + b*sqrt(c + d*x))**(15/2)/(15*b**8*d**4) +
12*(a + b*sqrt(c + d*x))**(11/2)*(7*a**2 - b**2*c)/(11*b**8*d**4) + 4*(a + b*sq
rt(c + d*x))**(7/2)*(35*a**4 - 30*a**2*b**2*c + 3*b**4*c**2)/(7*b**8*d**4) + 4*(
a + b*sqrt(c + d*x))**(3/2)*(a**2 - b**2*c)**2*(7*a**2 - b**2*c)/(3*b**8*d**4)

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Mathematica [A]  time = 0.486199, size = 285, normalized size = 0.88 \[ \frac{4 \left (-\frac{5}{9} \left (7 a^3-3 a b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}+\frac{3}{11} \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}+\frac{1}{3} \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}-a \left (a^2-b^2 c\right )^3 \sqrt{a+b \sqrt{c+d x}}-\frac{3}{5} \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}+\frac{1}{7} \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}+\frac{1}{15} \left (a+b \sqrt{c+d x}\right )^{15/2}-\frac{7}{13} a \left (a+b \sqrt{c+d x}\right )^{13/2}\right )}{b^8 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(4*(-(a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]]) + ((a^2 - b^2*c)^2*(7*a^2 - b
^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/3 - (3*(7*a^5 - 10*a^3*b^2*c + 3*a*b^4*c^2)*(
a + b*Sqrt[c + d*x])^(5/2))/5 + ((35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt
[c + d*x])^(7/2))/7 - (5*(7*a^3 - 3*a*b^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/9 + (3
*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/11 - (7*a*(a + b*Sqrt[c + d*x])^(
13/2))/13 + (a + b*Sqrt[c + d*x])^(15/2)/15))/(b^8*d^4)

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Maple [A]  time = 0.004, size = 383, normalized size = 1.2 \[ 4\,{\frac{1}{{d}^{4}{b}^{8}} \left ( 1/15\, \left ( a+b\sqrt{dx+c} \right ) ^{15/2}-{\frac{7\,a \left ( a+b\sqrt{dx+c} \right ) ^{13/2}}{13}}+1/11\, \left ( -3\,{b}^{2}c+21\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{11/2}+1/9\, \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) - \left ( -3\,{b}^{2}c+15\,{a}^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{9/2}+1/7\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}- \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{7/2}+1/5\, \left ( -6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}a- \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}+1/3\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}+6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{3/2}- \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}a\sqrt{a+b\sqrt{dx+c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/d^4/b^8*(1/15*(a+b*(d*x+c)^(1/2))^(15/2)-7/13*a*(a+b*(d*x+c)^(1/2))^(13/2)+1/1
1*(-3*b^2*c+21*a^2)*(a+b*(d*x+c)^(1/2))^(11/2)+1/9*(-8*(-b^2*c+a^2)*a-2*a*(-2*b^
2*c+6*a^2)-(-3*b^2*c+15*a^2)*a)*(a+b*(d*x+c)^(1/2))^(9/2)+1/7*((-b^2*c+a^2)*(-2*
b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+a^2)^2-(-8*(-b^2*c+a^2)*a-2*a*(-2*b^2*c+
6*a^2))*a)*(a+b*(d*x+c)^(1/2))^(7/2)+1/5*(-6*(-b^2*c+a^2)^2*a-((-b^2*c+a^2)*(-2*
b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+a^2)^2)*a)*(a+b*(d*x+c)^(1/2))^(5/2)+1/3
*((-b^2*c+a^2)^3+6*(-b^2*c+a^2)^2*a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-(-b^2*c+a^2)^3*
a*(a+b*(d*x+c)^(1/2))^(1/2))

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Maxima [A]  time = 0.69431, size = 362, normalized size = 1.12 \[ \frac{4 \,{\left (3003 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{15}{2}} - 24255 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{13}{2}} a - 12285 \,{\left (b^{2} c - 7 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} + 25025 \,{\left (3 \, a b^{2} c - 7 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} + 6435 \,{\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} - 27027 \,{\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} - 15015 \,{\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} + 45045 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt{\sqrt{d x + c} b + a}\right )}}{45045 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(sqrt(d*x + c)*b + a),x, algorithm="maxima")

[Out]

4/45045*(3003*(sqrt(d*x + c)*b + a)^(15/2) - 24255*(sqrt(d*x + c)*b + a)^(13/2)*
a - 12285*(b^2*c - 7*a^2)*(sqrt(d*x + c)*b + a)^(11/2) + 25025*(3*a*b^2*c - 7*a^
3)*(sqrt(d*x + c)*b + a)^(9/2) + 6435*(3*b^4*c^2 - 30*a^2*b^2*c + 35*a^4)*(sqrt(
d*x + c)*b + a)^(7/2) - 27027*(3*a*b^4*c^2 - 10*a^3*b^2*c + 7*a^5)*(sqrt(d*x + c
)*b + a)^(5/2) - 15015*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*(sqrt(d*
x + c)*b + a)^(3/2) + 45045*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*sqrt
(sqrt(d*x + c)*b + a))/(b^8*d^4)

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Fricas [A]  time = 0.33755, size = 312, normalized size = 0.96 \[ -\frac{4 \,{\left (3234 \, a b^{6} d^{3} x^{3} - 17280 \, a b^{6} c^{3} + 46976 \, a^{3} b^{4} c^{2} - 44544 \, a^{5} b^{2} c + 14336 \, a^{7} - 28 \,{\left (141 \, a b^{6} c - 140 \, a^{3} b^{4}\right )} d^{2} x^{2} + 64 \,{\left (87 \, a b^{6} c^{2} - 170 \, a^{3} b^{4} c + 84 \, a^{5} b^{2}\right )} d x -{\left (3003 \, b^{7} d^{3} x^{3} - 4992 \, b^{7} c^{3} + 18816 \, a^{2} b^{5} c^{2} - 20480 \, a^{4} b^{3} c + 7168 \, a^{6} b - 252 \,{\left (13 \, b^{7} c - 14 \, a^{2} b^{5}\right )} d^{2} x^{2} + 32 \,{\left (117 \, b^{7} c^{2} - 267 \, a^{2} b^{5} c + 140 \, a^{4} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{45045 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/45045*(3234*a*b^6*d^3*x^3 - 17280*a*b^6*c^3 + 46976*a^3*b^4*c^2 - 44544*a^5*b
^2*c + 14336*a^7 - 28*(141*a*b^6*c - 140*a^3*b^4)*d^2*x^2 + 64*(87*a*b^6*c^2 - 1
70*a^3*b^4*c + 84*a^5*b^2)*d*x - (3003*b^7*d^3*x^3 - 4992*b^7*c^3 + 18816*a^2*b^
5*c^2 - 20480*a^4*b^3*c + 7168*a^6*b - 252*(13*b^7*c - 14*a^2*b^5)*d^2*x^2 + 32*
(117*b^7*c^2 - 267*a^2*b^5*c + 140*a^4*b^3)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x +
c)*b + a)/(b^8*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{a + b \sqrt{c + d x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b*(d*x+c)**(1/2))**(1/2),x)

[Out]

Integral(x**3/sqrt(a + b*sqrt(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.338504, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done