Optimal. Leaf size=375 \[ -\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac {24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac {4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]
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Rubi [A] time = 0.37, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6689, 43} \begin {gather*} \frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}+\frac {8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac {24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac {4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 6689
Rubi steps
\begin {align*} \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx &=\frac {\int \left (a \left (1+\frac {3 c}{a}\right ) x^2 \sqrt {a+b x}+4 b x^3 \sqrt {a+b x}-3 a \left (1+\frac {c}{3 a}\right ) x^2 \sqrt {c+b x}-4 b x^3 \sqrt {c+b x}\right ) \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int x^3 \sqrt {a+b x} \, dx}{(a-c)^3}-\frac {(4 b) \int x^3 \sqrt {c+b x} \, dx}{(a-c)^3}-\frac {(3 a+c) \int x^2 \sqrt {c+b x} \, dx}{(a-c)^3}+\frac {(a+3 c) \int x^2 \sqrt {a+b x} \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac {(4 b) \int \left (-\frac {c^3 \sqrt {c+b x}}{b^3}+\frac {3 c^2 (c+b x)^{3/2}}{b^3}-\frac {3 c (c+b x)^{5/2}}{b^3}+\frac {(c+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac {(3 a+c) \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-c)^3}+\frac {(a+3 c) \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-c)^3}\\ &=-\frac {8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac {24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac {24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac {8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac {8 c^3 (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {2 c^2 (3 a+c) (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac {24 c^2 (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {4 c (3 a+c) (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac {24 c (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac {8 (c+b x)^{9/2}}{9 b^3 (a-c)^3}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 282, normalized size = 0.75 \begin {gather*} \frac {2 \left (-40 a^4 \sqrt {a+b x}+4 a^3 \sqrt {a+b x} (5 b x+18 c)-3 a^2 b x \sqrt {a+b x} (5 b x+12 c)+a \left (5 b^3 x^3 \left (13 \sqrt {a+b x}-27 \sqrt {b x+c}\right )+27 b^2 c x^2 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )-72 c^3 \sqrt {b x+c}+36 b c^2 x \sqrt {b x+c}\right )+5 \left (28 b^4 x^4 \left (\sqrt {a+b x}-\sqrt {b x+c}\right )+b^3 c x^3 \left (27 \sqrt {a+b x}-13 \sqrt {b x+c}\right )+3 b^2 c^2 x^2 \sqrt {b x+c}+8 c^4 \sqrt {b x+c}-4 b c^3 x \sqrt {b x+c}\right )\right )}{315 b^3 (a-c)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.82, size = 267, normalized size = 0.71 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (40 a^4-20 a^3 (b x+c)-52 a^3 c+15 a^2 (b x+c)^2+6 a^2 c (b x+c)-21 a^2 c^2-141 a c^2 (b x+c)-65 a (b x+c)^3+168 a c (b x+c)^2+38 a c^3+155 c^3 (b x+c)-435 c^2 (b x+c)^2-140 (b x+c)^4+425 c (b x+c)^3-5 c^4\right )}{315 b^3 (a-c)^3}-\frac {2 \left (315 a c^2 (b x+c)^{3/2}+135 a (b x+c)^{7/2}-378 a c (b x+c)^{5/2}-315 c^3 (b x+c)^{3/2}+630 c^2 (b x+c)^{5/2}+140 (b x+c)^{9/2}-495 c (b x+c)^{7/2}\right )}{315 b^3 (a-c)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 208, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left ({\left (140 \, b^{4} x^{4} - 40 \, a^{4} + 72 \, a^{3} c + 5 \, {\left (13 \, a b^{3} + 27 \, b^{3} c\right )} x^{3} - 3 \, {\left (5 \, a^{2} b^{2} - 9 \, a b^{2} c\right )} x^{2} + 4 \, {\left (5 \, a^{3} b - 9 \, a^{2} b c\right )} x\right )} \sqrt {b x + a} - {\left (140 \, b^{4} x^{4} + 72 \, a c^{3} - 40 \, c^{4} + 5 \, {\left (27 \, a b^{3} + 13 \, b^{3} c\right )} x^{3} + 3 \, {\left (9 \, a b^{2} c - 5 \, b^{2} c^{2}\right )} x^{2} - 4 \, {\left (9 \, a b c^{2} - 5 \, b c^{3}\right )} x\right )} \sqrt {b x + c}\right )}}{315 \, {\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.81, size = 1447, normalized size = 3.86
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 294, normalized size = 0.78 \begin {gather*} \frac {2 \left (\frac {\left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}\right ) a}{\left (a -c \right )^{3} b^{3}}-\frac {6 \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 ) a}{\left (a -c \right )^{3} b^{3}}+\frac {6 \left (\frac {\left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}\right ) c}{\left (a -c \right )^{3} 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 ) c}{\left (a -c \right )^{3} b^{3}}+\frac {-\frac {8 \left (b x +a \right )^{\frac {3}{2}} a^{3}}{3}+\frac {24 \left (b x +a \right )^{\frac {5}{2}} a^{2}}{5}-\frac {24 \left (b x +a \right )^{\frac {7}{2}} a}{7}+\frac {8 \left (b x +a \right )^{\frac {9}{2}}}{9}}{\left (a -c \right )^{3} b^{3}}-\frac {8 \left (-\frac {\left (b x +c \right )^{\frac {3}{2}} c^{3}}{3}+\frac {3 \left (b x +c \right )^{\frac {5}{2}} c^{2}}{5}-\frac {3 \left (b x +c \right )^{\frac {7}{2}} c}{7}+\frac {\left (b x +c \right )^{\frac {9}{2}}}{9}\right )}{\left (a -c \right )^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.30, size = 529, normalized size = 1.41 \begin {gather*} \frac {x^3\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{7\,b}-\frac {x^3\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{7\,b}-\frac {8\,c^2\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{15\,b^3}-\frac {x^2\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{5\,b}+\frac {8\,a^2\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{15\,b^3}+\frac {x^2\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{5\,b}+\frac {8\,b\,x^4\,\sqrt {a+b\,x}}{9\,{\left (a-c\right )}^3}-\frac {8\,b\,x^4\,\sqrt {c+b\,x}}{9\,{\left (a-c\right )}^3}+\frac {4\,c\,x\,\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {6\,c\,\left (\frac {64\,b\,c}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {c+b\,x}}{15\,b^2}-\frac {4\,a\,x\,\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {6\,a\,\left (\frac {64\,a\,b}{9\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{7\,b}\right )\,\sqrt {a+b\,x}}{15\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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