3.401 \(\int \frac {x^2}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.13, 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 = {2104, 43} \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac {2 c^2 (b x+c)^{3/2}}{3 b^3 (a-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (a-c)}-\frac {2 (b x+c)^{7/2}}{7 b^3 (a-c)}+\frac {4 c (b x+c)^{5/2}}{5 b^3 (a-c)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x]),x]

[Out]

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

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 2104

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

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 140, normalized size = 0.95 \[ \frac {2 \left (8 a^3 \sqrt {a+b x}-4 a^2 b x \sqrt {a+b x}+15 b^3 x^3 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )+3 a b^2 x^2 \sqrt {a+b x}-3 b^2 c x^2 \sqrt {b x+c}-8 c^3 \sqrt {b x+c}+4 b c^2 x \sqrt {b x+c}\right )}{105 b^3 (a-c)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x]),x]

[Out]

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

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fricas [A]  time = 0.44, size = 94, normalized size = 0.64 \[ \frac {2 \, {\left ({\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a} - {\left (15 \, b^{3} x^{3} + 3 \, b^{2} c x^{2} - 4 \, b c^{2} x + 8 \, c^{3}\right )} \sqrt {b x + c}\right )}}{105 \, {\left (a b^{3} - b^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.27, size = 390, normalized size = 2.65 \[ -\frac {2}{105} \, {\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {5 \, {\left (a^{2} b^{9} - 2 \, a b^{9} c + b^{9} c^{2}\right )} {\left (b x + a\right )}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}} - \frac {15 \, a^{3} b^{9} - 31 \, a^{2} b^{9} c + 17 \, a b^{9} c^{2} - b^{9} c^{3}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} + \frac {45 \, a^{4} b^{9} - 96 \, a^{3} b^{9} c + 53 \, a^{2} b^{9} c^{2} + 2 \, a b^{9} c^{3} - 4 \, b^{9} c^{4}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} {\left (b x + a\right )} - \frac {15 \, a^{5} b^{9} - 33 \, a^{4} b^{9} c + 17 \, a^{3} b^{9} c^{2} - 3 \, a^{2} b^{9} c^{3} + 12 \, a b^{9} c^{4} - 8 \, b^{9} c^{5}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} \sqrt {b x + c} + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} - 42 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, {\left (a b^{3} - b^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*((3*(b*x + a)*(5*(a^2*b^9 - 2*a*b^9*c + b^9*c^2)*(b*x + a)/(a^3*b^12 - 3*a^2*b^12*c + 3*a*b^12*c^2 - b^
12*c^3) - (15*a^3*b^9 - 31*a^2*b^9*c + 17*a*b^9*c^2 - b^9*c^3)/(a^3*b^12 - 3*a^2*b^12*c + 3*a*b^12*c^2 - b^12*
c^3)) + (45*a^4*b^9 - 96*a^3*b^9*c + 53*a^2*b^9*c^2 + 2*a*b^9*c^3 - 4*b^9*c^4)/(a^3*b^12 - 3*a^2*b^12*c + 3*a*
b^12*c^2 - b^12*c^3))*(b*x + a) - (15*a^5*b^9 - 33*a^4*b^9*c + 17*a^3*b^9*c^2 - 3*a^2*b^9*c^3 + 12*a*b^9*c^4 -
8*b^9*c^5)/(a^3*b^12 - 3*a^2*b^12*c + 3*a*b^12*c^2 - b^12*c^3))*sqrt(b*x + c) + 2/105*(15*(b*x + a)^(7/2) - 4
2*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2)/(a*b^3 - b^3*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {b x + a} + \sqrt {b x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c)), x)

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mupad [B]  time = 2.95, size = 179, normalized size = 1.22 \[ \frac {2\,x^3\,\sqrt {a+b\,x}}{7\,\left (a-c\right )}-\frac {2\,x^3\,\sqrt {c+b\,x}}{7\,\left (a-c\right )}+\frac {16\,a^3\,\sqrt {a+b\,x}}{105\,b^3\,\left (a-c\right )}-\frac {16\,c^3\,\sqrt {c+b\,x}}{105\,b^3\,\left (a-c\right )}+\frac {2\,a\,x^2\,\sqrt {a+b\,x}}{35\,b\,\left (a-c\right )}-\frac {8\,a^2\,x\,\sqrt {a+b\,x}}{105\,b^2\,\left (a-c\right )}-\frac {2\,c\,x^2\,\sqrt {c+b\,x}}{35\,b\,\left (a-c\right )}+\frac {8\,c^2\,x\,\sqrt {c+b\,x}}{105\,b^2\,\left (a-c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {a + b x} + \sqrt {b x + c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(a + b*x) + sqrt(b*x + c)), x)

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