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

Optimal. Leaf size=326 \[ \frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac{4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4} \]

[Out]

(-4*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) + (4*(a^2 - b^2*c
)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*
b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*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])^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 -
3*b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a +
b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) - (28*a*(a + b*Sqrt[c + d*x])^(15/2))/(15*
b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(17/2))/(17*b^8*d^4)

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Rubi [A]  time = 0.530122, antiderivative size = 326, 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 \[ \frac{12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac{20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac{12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac{4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac{4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac{4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac{28 a \left (a+b \sqrt{c+d x}\right )^{15/2}}{15 b^8 d^4} \]

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*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) + (4*(a^2 - b^2*c
)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*
b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*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])^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 -
3*b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a +
b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) - (28*a*(a + b*Sqrt[c + d*x])^(15/2))/(15*
b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(17/2))/(17*b^8*d^4)

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Rubi in Sympy [A]  time = 32.3162, size = 306, normalized size = 0.94 \[ - \frac{28 a \left (a + b \sqrt{c + d x}\right )^{\frac{15}{2}}}{15 b^{8} d^{4}} - \frac{20 a \left (a + b \sqrt{c + d x}\right )^{\frac{11}{2}} \left (7 a^{2} - 3 b^{2} c\right )}{11 b^{8} d^{4}} - \frac{12 a \left (a + b \sqrt{c + d x}\right )^{\frac{7}{2}} \left (a^{2} - b^{2} c\right ) \left (7 a^{2} - 3 b^{2} c\right )}{7 b^{8} d^{4}} - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{3}}{3 b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{17}{2}}}{17 b^{8} d^{4}} + \frac{12 \left (a + b \sqrt{c + d x}\right )^{\frac{13}{2}} \left (7 a^{2} - b^{2} c\right )}{13 b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{9}{2}} \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{9 b^{8} d^{4}} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}} \left (a^{2} - b^{2} c\right )^{2} \left (7 a^{2} - b^{2} c\right )}{5 b^{8} d^{4}} \]

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))**(15/2)/(15*b**8*d**4) - 20*a*(a + b*sqrt(c + d*x))*
*(11/2)*(7*a**2 - 3*b**2*c)/(11*b**8*d**4) - 12*a*(a + b*sqrt(c + d*x))**(7/2)*(
a**2 - b**2*c)*(7*a**2 - 3*b**2*c)/(7*b**8*d**4) - 4*a*(a + b*sqrt(c + d*x))**(3
/2)*(a**2 - b**2*c)**3/(3*b**8*d**4) + 4*(a + b*sqrt(c + d*x))**(17/2)/(17*b**8*
d**4) + 12*(a + b*sqrt(c + d*x))**(13/2)*(7*a**2 - b**2*c)/(13*b**8*d**4) + 4*(a
+ b*sqrt(c + d*x))**(9/2)*(35*a**4 - 30*a**2*b**2*c + 3*b**4*c**2)/(9*b**8*d**4
) + 4*(a + b*sqrt(c + d*x))**(5/2)*(a**2 - b**2*c)**2*(7*a**2 - b**2*c)/(5*b**8*
d**4)

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Mathematica [A]  time = 0.290161, size = 232, normalized size = 0.71 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2} \left (-14336 a^7+21504 a^6 b \sqrt{c+d x}+3840 a^5 b^2 (10 c-7 d x)-640 a^4 b^3 (104 c-49 d x) \sqrt{c+d x}-48 a^3 b^4 \left (616 c^2-1080 c d x+735 d^2 x^2\right )+24 a^2 b^5 \sqrt{c+d x} \left (2960 c^2-2716 c d x+1617 d^2 x^2\right )+6 a b^6 \left (320 c^3-3936 c^2 d x+5754 c d^2 x^2-7007 d^3 x^3\right )-231 b^7 \sqrt{c+d x} \left (128 c^3-160 c^2 d x+180 c d^2 x^2-195 d^3 x^3\right )\right )}{765765 b^8 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(4*(a + b*Sqrt[c + d*x])^(3/2)*(-14336*a^7 + 3840*a^5*b^2*(10*c - 7*d*x) + 21504
*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3*(104*c - 49*d*x)*Sqrt[c + d*x] - 48*a^3*b^4*(
616*c^2 - 1080*c*d*x + 735*d^2*x^2) + 24*a^2*b^5*Sqrt[c + d*x]*(2960*c^2 - 2716*
c*d*x + 1617*d^2*x^2) + 6*a*b^6*(320*c^3 - 3936*c^2*d*x + 5754*c*d^2*x^2 - 7007*
d^3*x^3) - 231*b^7*Sqrt[c + d*x]*(128*c^3 - 160*c^2*d*x + 180*c*d^2*x^2 - 195*d^
3*x^3)))/(765765*b^8*d^4)

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Maple [A]  time = 0.006, size = 383, normalized size = 1.2 \[ 4\,{\frac{1}{{d}^{4}{b}^{8}} \left ( 1/17\, \left ( a+b\sqrt{dx+c} \right ) ^{17/2}-{\frac{7\,a \left ( a+b\sqrt{dx+c} \right ) ^{15/2}}{15}}+1/13\, \left ( -3\,{b}^{2}c+21\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{13/2}+1/11\, \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) - \left ( -3\,{b}^{2}c+15\,{a}^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{11/2}+1/9\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}- \left ( -8\, \left ( -{b}^{2}c+{a}^{2} \right ) a-2\,a \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{9/2}+1/7\, \left ( -6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}a- \left ( \left ( -{b}^{2}c+{a}^{2} \right ) \left ( -2\,{b}^{2}c+6\,{a}^{2} \right ) +8\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) + \left ( -{b}^{2}c+{a}^{2} \right ) ^{2} \right ) a \right ) \left ( a+b\sqrt{dx+c} \right ) ^{7/2}+1/5\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}+6\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}a \left ( a+b\sqrt{dx+c} \right ) ^{3/2} \right ) } \]

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*(1/17*(a+b*(d*x+c)^(1/2))^(17/2)-7/15*a*(a+b*(d*x+c)^(1/2))^(15/2)+1/1
3*(-3*b^2*c+21*a^2)*(a+b*(d*x+c)^(1/2))^(13/2)+1/11*(-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))^(11/2)+1/9*((-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))^(9/2)+1/7*(-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))^(7/2)+1
/5*((-b^2*c+a^2)^3+6*(-b^2*c+a^2)^2*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)-1/3*(-b^2*c+a
^2)^3*a*(a+b*(d*x+c)^(1/2))^(3/2))

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Maxima [A]  time = 0.725595, size = 362, normalized size = 1.11 \[ \frac{4 \,{\left (45045 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{17}{2}} - 357357 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{15}{2}} a - 176715 \,{\left (b^{2} c - 7 \, a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{13}{2}} + 348075 \,{\left (3 \, a b^{2} c - 7 \, a^{3}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{11}{2}} + 85085 \,{\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{9}{2}} - 328185 \,{\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{7}{2}} - 153153 \,{\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}} + 255255 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}\right )}}{765765 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/765765*(45045*(sqrt(d*x + c)*b + a)^(17/2) - 357357*(sqrt(d*x + c)*b + a)^(15/
2)*a - 176715*(b^2*c - 7*a^2)*(sqrt(d*x + c)*b + a)^(13/2) + 348075*(3*a*b^2*c -
7*a^3)*(sqrt(d*x + c)*b + a)^(11/2) + 85085*(3*b^4*c^2 - 30*a^2*b^2*c + 35*a^4)
*(sqrt(d*x + c)*b + a)^(9/2) - 328185*(3*a*b^4*c^2 - 10*a^3*b^2*c + 7*a^5)*(sqrt
(d*x + c)*b + a)^(7/2) - 153153*(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)^(5/2) + 255255*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c -
a^7)*(sqrt(d*x + c)*b + a)^(3/2))/(b^8*d^4)

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Fricas [A]  time = 0.346331, size = 386, normalized size = 1.18 \[ \frac{4 \,{\left (45045 \, b^{8} d^{4} x^{4} - 29568 \, b^{8} c^{4} + 72960 \, a^{2} b^{6} c^{3} - 96128 \, a^{4} b^{4} c^{2} + 59904 \, a^{6} b^{2} c - 14336 \, a^{8} + 231 \,{\left (15 \, b^{8} c - 14 \, a^{2} b^{6}\right )} d^{3} x^{3} - 28 \,{\left (165 \, b^{8} c^{2} - 291 \, a^{2} b^{6} c + 140 \, a^{4} b^{4}\right )} d^{2} x^{2} + 32 \,{\left (231 \, b^{8} c^{3} - 555 \, a^{2} b^{6} c^{2} + 520 \, a^{4} b^{4} c - 168 \, a^{6} b^{2}\right )} d x +{\left (3003 \, a b^{7} d^{3} x^{3} - 27648 \, a b^{7} c^{3} + 41472 \, a^{3} b^{5} c^{2} - 28160 \, a^{5} b^{3} c + 7168 \, a^{7} b - 3528 \,{\left (2 \, a b^{7} c - a^{3} b^{5}\right )} d^{2} x^{2} + 32 \,{\left (417 \, a b^{7} c^{2} - 417 \, a^{3} b^{5} c + 140 \, a^{5} b^{3}\right )} d x\right )} \sqrt{d x + c}\right )} \sqrt{\sqrt{d x + c} b + a}}{765765 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/765765*(45045*b^8*d^4*x^4 - 29568*b^8*c^4 + 72960*a^2*b^6*c^3 - 96128*a^4*b^4*
c^2 + 59904*a^6*b^2*c - 14336*a^8 + 231*(15*b^8*c - 14*a^2*b^6)*d^3*x^3 - 28*(16
5*b^8*c^2 - 291*a^2*b^6*c + 140*a^4*b^4)*d^2*x^2 + 32*(231*b^8*c^3 - 555*a^2*b^6
*c^2 + 520*a^4*b^4*c - 168*a^6*b^2)*d*x + (3003*a*b^7*d^3*x^3 - 27648*a*b^7*c^3
+ 41472*a^3*b^5*c^2 - 28160*a^5*b^3*c + 7168*a^7*b - 3528*(2*a*b^7*c - a^3*b^5)*
d^2*x^2 + 32*(417*a*b^7*c^2 - 417*a^3*b^5*c + 140*a^5*b^3)*d*x)*sqrt(d*x + c))*s
qrt(sqrt(d*x + c)*b + a)/(b^8*d^4)

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

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 = 0.375332, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done