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

Optimal. Leaf size=147 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2103, 43} \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2103

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[c/(e*(b*c - a*d)
), Int[(u*Sqrt[a + b*x])/x, x], x] - Dist[a/(f*(b*c - a*d)), Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {a+b x}+\sqrt {a+c x}} \, dx &=\frac {\int x^2 \sqrt {a+b x} \, dx}{b-c}-\frac {\int x^2 \sqrt {a+c x} \, dx}{b-c}\\ &=\frac {\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx}{b-c}-\frac {\int \left (\frac {a^2 \sqrt {a+c x}}{c^2}-\frac {2 a (a+c x)^{3/2}}{c^2}+\frac {(a+c x)^{5/2}}{c^2}\right ) \, dx}{b-c}\\ &=\frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 (b-c) c^3}+\frac {4 a (a+c x)^{5/2}}{5 (b-c) c^3}-\frac {2 (a+c x)^{7/2}}{7 (b-c) c^3}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 147, normalized size = 1.00 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

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

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fricas [A]  time = 0.45, size = 122, normalized size = 0.83 \[ \frac {2 \, {\left ({\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{2} c^{3} x^{2} - 4 \, a^{2} b c^{3} x + 8 \, a^{3} c^{3}\right )} \sqrt {b x + a} - {\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{3} c^{2} x^{2} - 4 \, a^{2} b^{3} c x + 8 \, a^{3} b^{3}\right )} \sqrt {c x + a}\right )}}{105 \, {\left (b^{4} c^{3} - b^{3} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x, algorithm="fricas")

[Out]

2/105*((15*b^3*c^3*x^3 + 3*a*b^2*c^3*x^2 - 4*a^2*b*c^3*x + 8*a^3*c^3)*sqrt(b*x + a) - (15*b^3*c^3*x^3 + 3*a*b^
3*c^2*x^2 - 4*a^2*b^3*c*x + 8*a^3*b^3)*sqrt(c*x + a))/(b^4*c^3 - b^3*c^4)

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giac [B]  time = 0.43, size = 451, normalized size = 3.07 \[ -\frac {2}{105} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {5 \, {\left (b^{17} c^{5} {\left | b \right |} - 2 \, b^{16} c^{6} {\left | b \right |} + b^{15} c^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}} + \frac {a b^{18} c^{4} {\left | b \right |} - 17 \, a b^{17} c^{5} {\left | b \right |} + 31 \, a b^{16} c^{6} {\left | b \right |} - 15 \, a b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} - \frac {4 \, a^{2} b^{19} c^{3} {\left | b \right |} - 2 \, a^{2} b^{18} c^{4} {\left | b \right |} - 53 \, a^{2} b^{17} c^{5} {\left | b \right |} + 96 \, a^{2} b^{16} c^{6} {\left | b \right |} - 45 \, a^{2} b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} {\left (b x + a\right )} + \frac {8 \, a^{3} b^{20} c^{2} {\left | b \right |} - 12 \, a^{3} b^{19} c^{3} {\left | b \right |} + 3 \, a^{3} b^{18} c^{4} {\left | b \right |} - 17 \, a^{3} b^{17} c^{5} {\left | b \right |} + 33 \, a^{3} b^{16} c^{6} {\left | b \right |} - 15 \, a^{3} b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} - 42 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, {\left (b^{4} - b^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x, algorithm="giac")

[Out]

-2/105*sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*((3*(b*x + a)*(5*(b^17*c^5*abs(b) - 2*b^16*c^6*abs(b) + b^15*c^7*ab
s(b))*(b*x + a)/(b^23*c^5 - 3*b^22*c^6 + 3*b^21*c^7 - b^20*c^8) + (a*b^18*c^4*abs(b) - 17*a*b^17*c^5*abs(b) +
31*a*b^16*c^6*abs(b) - 15*a*b^15*c^7*abs(b))/(b^23*c^5 - 3*b^22*c^6 + 3*b^21*c^7 - b^20*c^8)) - (4*a^2*b^19*c^
3*abs(b) - 2*a^2*b^18*c^4*abs(b) - 53*a^2*b^17*c^5*abs(b) + 96*a^2*b^16*c^6*abs(b) - 45*a^2*b^15*c^7*abs(b))/(
b^23*c^5 - 3*b^22*c^6 + 3*b^21*c^7 - b^20*c^8))*(b*x + a) + (8*a^3*b^20*c^2*abs(b) - 12*a^3*b^19*c^3*abs(b) +
3*a^3*b^18*c^4*abs(b) - 17*a^3*b^17*c^5*abs(b) + 33*a^3*b^16*c^6*abs(b) - 15*a^3*b^15*c^7*abs(b))/(b^23*c^5 -
3*b^22*c^6 + 3*b^21*c^7 - b^20*c^8)) + 2/105*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a
^2)/(b^4 - b^3*c)

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maple [A]  time = 0.00, size = 90, normalized size = 0.61 \[ \frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {4 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}}{\left (b -c \right ) b^{3}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (c x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (c x +a \right )^{\frac {7}{2}}}{7}\right )}{\left (b -c \right ) c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {b x + a} + \sqrt {c x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x, algorithm="maxima")

[Out]

integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a)), x)

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mupad [B]  time = 2.92, size = 179, normalized size = 1.22 \[ \frac {2\,x^3\,\sqrt {a+b\,x}}{7\,\left (b-c\right )}-\frac {2\,x^3\,\sqrt {a+c\,x}}{7\,\left (b-c\right )}+\frac {16\,a^3\,\sqrt {a+b\,x}}{105\,b^3\,\left (b-c\right )}-\frac {16\,a^3\,\sqrt {a+c\,x}}{105\,c^3\,\left (b-c\right )}+\frac {2\,a\,x^2\,\sqrt {a+b\,x}}{35\,b\,\left (b-c\right )}-\frac {8\,a^2\,x\,\sqrt {a+b\,x}}{105\,b^2\,\left (b-c\right )}-\frac {2\,a\,x^2\,\sqrt {a+c\,x}}{35\,c\,\left (b-c\right )}+\frac {8\,a^2\,x\,\sqrt {a+c\,x}}{105\,c^2\,\left (b-c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x)^(1/2) + (a + c*x)^(1/2)),x)

[Out]

(2*x^3*(a + b*x)^(1/2))/(7*(b - c)) - (2*x^3*(a + c*x)^(1/2))/(7*(b - c)) + (16*a^3*(a + b*x)^(1/2))/(105*b^3*
(b - c)) - (16*a^3*(a + c*x)^(1/2))/(105*c^3*(b - c)) + (2*a*x^2*(a + b*x)^(1/2))/(35*b*(b - c)) - (8*a^2*x*(a
+ b*x)^(1/2))/(105*b^2*(b - c)) - (2*a*x^2*(a + c*x)^(1/2))/(35*c*(b - c)) + (8*a^2*x*(a + c*x)^(1/2))/(105*c
^2*(b - c))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2)),x)

[Out]

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

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