Optimal. Leaf size=277 \[ -\frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac {2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}+\frac {8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}+\frac {8 a^2 (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac {2 a^2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^3}-\frac {8 a (a+c x)^{5/2}}{5 (b-c)^3 c^2}+\frac {4 a (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^3}-\frac {2 (3 b+c) (a+c x)^{7/2}}{7 (b-c)^3 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 277, 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 = {6822, 45}
\begin {gather*} \frac {2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac {2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac {8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac {4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac {8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac {4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac {8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6822
Rubi steps
\begin {align*} \int \frac {x^4}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx &=\frac {\int \left (4 a x \sqrt {a+b x}+b \left (1+\frac {3 c}{b}\right ) x^2 \sqrt {a+b x}-4 a x \sqrt {a+c x}-3 b \left (1+\frac {c}{3 b}\right ) x^2 \sqrt {a+c x}\right ) \, dx}{(b-c)^3}\\ &=\frac {(4 a) \int x \sqrt {a+b x} \, dx}{(b-c)^3}-\frac {(4 a) \int x \sqrt {a+c x} \, dx}{(b-c)^3}-\frac {(3 b+c) \int x^2 \sqrt {a+c x} \, dx}{(b-c)^3}+\frac {(b+3 c) \int x^2 \sqrt {a+b x} \, dx}{(b-c)^3}\\ &=\frac {(4 a) \int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{(b-c)^3}-\frac {(4 a) \int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx}{(b-c)^3}-\frac {(3 b+c) \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)^3}+\frac {(b+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}{(b-c)^3}\\ &=-\frac {8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}+\frac {2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}+\frac {8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac {4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac {2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}+\frac {8 a^2 (a+c x)^{3/2}}{3 (b-c)^3 c^2}-\frac {2 a^2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c^3}-\frac {8 a (a+c x)^{5/2}}{5 (b-c)^3 c^2}+\frac {4 a (3 b+c) (a+c x)^{5/2}}{5 (b-c)^3 c^3}-\frac {2 (3 b+c) (a+c x)^{7/2}}{7 (b-c)^3 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 10.27, size = 114, normalized size = 0.41 \begin {gather*} -\frac {2 \left (b^3 (a+c x)^{3/2} \left (8 a^2 (b-2 c)-12 a (b-2 c) c x+5 c^2 (3 b+c) x^2\right )+c^3 (a+b x)^{3/2} \left (8 a^2 (2 b-c)+12 a b (-2 b+c) x-5 b^2 (b+3 c) x^2\right )\right )}{35 b^3 (b-c)^3 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 246, normalized size = 0.89
method | result | size |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (b -c \right )^{3} b^{2}}+\frac {8 a \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (b x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right )^{3} b^{2}}-\frac {8 a \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (c x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right )^{3} c^{2}}+\frac {6 c \left (\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} b^{3}}-\frac {6 b \left (\frac {\left (c x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (c x +a \right )^{\frac {5}{2}}}{5}+\frac {a^{2} \left (c x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} c^{3}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (c x +a \right )^{\frac {5}{2}}}{5}+\frac {a^{2} \left (c x +a \right )^{\frac {3}{2}}}{3}\right )}{\left (b -c \right )^{3} c^{2}}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 225, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left ({\left (16 \, a^{3} b c^{3} - 8 \, a^{3} c^{4} - 5 \, {\left (b^{4} c^{3} + 3 \, b^{3} c^{4}\right )} x^{3} - {\left (29 \, a b^{3} c^{3} + 3 \, a b^{2} c^{4}\right )} x^{2} - 4 \, {\left (2 \, a^{2} b^{2} c^{3} - a^{2} b c^{4}\right )} x\right )} \sqrt {b x + a} + {\left (8 \, a^{3} b^{4} - 16 \, a^{3} b^{3} c + 5 \, {\left (3 \, b^{4} c^{3} + b^{3} c^{4}\right )} x^{3} + {\left (3 \, a b^{4} c^{2} + 29 \, a b^{3} c^{3}\right )} x^{2} - 4 \, {\left (a^{2} b^{4} c - 2 \, a^{2} b^{3} c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{35 \, {\left (b^{6} c^{3} - 3 \, b^{5} c^{4} + 3 \, b^{4} c^{5} - b^{3} c^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 932 vs.
\(2 (237) = 474\).
time = 4.24, size = 932, normalized size = 3.36 \begin {gather*} -\frac {2}{35} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left ({\left (b x + a\right )} {\left (\frac {5 \, {\left (3 \, b^{22} c^{5} {\left | b \right |} - 17 \, b^{21} c^{6} {\left | b \right |} + 39 \, b^{20} c^{7} {\left | b \right |} - 45 \, b^{19} c^{8} {\left | b \right |} + 25 \, b^{18} c^{9} {\left | b \right |} - 3 \, b^{17} c^{10} {\left | b \right |} - 3 \, b^{16} c^{11} {\left | b \right |} + b^{15} c^{12} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{29} c^{5} - 9 \, b^{28} c^{6} + 36 \, b^{27} c^{7} - 84 \, b^{26} c^{8} + 126 \, b^{25} c^{9} - 126 \, b^{24} c^{10} + 84 \, b^{23} c^{11} - 36 \, b^{22} c^{12} + 9 \, b^{21} c^{13} - b^{20} c^{14}} + \frac {3 \, a b^{23} c^{4} {\left | b \right |} - 34 \, a b^{22} c^{5} {\left | b \right |} + 126 \, a b^{21} c^{6} {\left | b \right |} - 210 \, a b^{20} c^{7} {\left | b \right |} + 140 \, a b^{19} c^{8} {\left | b \right |} + 42 \, a b^{18} c^{9} {\left | b \right |} - 126 \, a b^{17} c^{10} {\left | b \right |} + 74 \, a b^{16} c^{11} {\left | b \right |} - 15 \, a b^{15} c^{12} {\left | b \right |}}{b^{29} c^{5} - 9 \, b^{28} c^{6} + 36 \, b^{27} c^{7} - 84 \, b^{26} c^{8} + 126 \, b^{25} c^{9} - 126 \, b^{24} c^{10} + 84 \, b^{23} c^{11} - 36 \, b^{22} c^{12} + 9 \, b^{21} c^{13} - b^{20} c^{14}}\right )} - \frac {4 \, a^{2} b^{24} c^{3} {\left | b \right |} - 26 \, a^{2} b^{23} c^{4} {\left | b \right |} + 85 \, a^{2} b^{22} c^{5} {\left | b \right |} - 203 \, a^{2} b^{21} c^{6} {\left | b \right |} + 385 \, a^{2} b^{20} c^{7} {\left | b \right |} - 539 \, a^{2} b^{19} c^{8} {\left | b \right |} + 511 \, a^{2} b^{18} c^{9} {\left | b \right |} - 305 \, a^{2} b^{17} c^{10} {\left | b \right |} + 103 \, a^{2} b^{16} c^{11} {\left | b \right |} - 15 \, a^{2} b^{15} c^{12} {\left | b \right |}}{b^{29} c^{5} - 9 \, b^{28} c^{6} + 36 \, b^{27} c^{7} - 84 \, b^{26} c^{8} + 126 \, b^{25} c^{9} - 126 \, b^{24} c^{10} + 84 \, b^{23} c^{11} - 36 \, b^{22} c^{12} + 9 \, b^{21} c^{13} - b^{20} c^{14}}\right )} {\left (b x + a\right )} + \frac {8 \, a^{3} b^{25} c^{2} {\left | b \right |} - 60 \, a^{3} b^{24} c^{3} {\left | b \right |} + 187 \, a^{3} b^{23} c^{4} {\left | b \right |} - 296 \, a^{3} b^{22} c^{5} {\left | b \right |} + 196 \, a^{3} b^{21} c^{6} {\left | b \right |} + 112 \, a^{3} b^{20} c^{7} {\left | b \right |} - 350 \, a^{3} b^{19} c^{8} {\left | b \right |} + 328 \, a^{3} b^{18} c^{9} {\left | b \right |} - 164 \, a^{3} b^{17} c^{10} {\left | b \right |} + 44 \, a^{3} b^{16} c^{11} {\left | b \right |} - 5 \, a^{3} b^{15} c^{12} {\left | b \right |}}{b^{29} c^{5} - 9 \, b^{28} c^{6} + 36 \, b^{27} c^{7} - 84 \, b^{26} c^{8} + 126 \, b^{25} c^{9} - 126 \, b^{24} c^{10} + 84 \, b^{23} c^{11} - 36 \, b^{22} c^{12} + 9 \, b^{21} c^{13} - b^{20} c^{14}}\right )} + \frac {2 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} b + 14 \, {\left (b x + a\right )}^{\frac {5}{2}} a b - 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b + 15 \, {\left (b x + a\right )}^{\frac {7}{2}} c - 42 \, {\left (b x + a\right )}^{\frac {5}{2}} a c + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} c\right )}}{35 \, {\left (b^{6} - 3 \, b^{5} c + 3 \, b^{4} c^{2} - b^{3} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.34, size = 429, normalized size = 1.55 \begin {gather*} \frac {x^2\,\left (\frac {12\,a\,\left (3\,b+c\right )}{7\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )\,\sqrt {a+c\,x}}{5\,c}-\frac {2\,a\,\left (\frac {8\,a^2}{{\left (b-c\right )}^3}-\frac {4\,a\,\left (\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}-\frac {12\,a\,\left (b^2+3\,c\,b\right )}{7\,b\,{\left (b-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b^2}+\frac {x\,\left (\frac {8\,a^2}{{\left (b-c\right )}^3}-\frac {4\,a\,\left (\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}-\frac {12\,a\,\left (b^2+3\,c\,b\right )}{7\,b\,{\left (b-c\right )}^3}\right )}{5\,b}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {2\,a\,\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {4\,a\,\left (\frac {12\,a\,\left (3\,b+c\right )}{7\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )}{5\,c}\right )\,\sqrt {a+c\,x}}{3\,c^2}+\frac {x^2\,\left (\frac {2\,a\,\left (5\,b+3\,c\right )}{{\left (b-c\right )}^3}-\frac {12\,a\,\left (b^2+3\,c\,b\right )}{7\,b\,{\left (b-c\right )}^3}\right )\,\sqrt {a+b\,x}}{5\,b}-\frac {x\,\left (\frac {8\,a^2}{{\left (b-c\right )}^3}+\frac {4\,a\,\left (\frac {12\,a\,\left (3\,b+c\right )}{7\,{\left (b-c\right )}^3}-\frac {2\,a\,\left (3\,b+5\,c\right )}{{\left (b-c\right )}^3}\right )}{5\,c}\right )\,\sqrt {a+c\,x}}{3\,c}-\frac {2\,x^3\,\left (3\,b+c\right )\,\sqrt {a+c\,x}}{7\,{\left (b-c\right )}^3}+\frac {2\,x^3\,\left (b^2+3\,c\,b\right )\,\sqrt {a+b\,x}}{7\,b\,{\left (b-c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________