Optimal. Leaf size=186 \[ -\frac {2 (B d-A e)}{3 e (b d-a e) (a+b x)^{3/2} (d+e x)^{3/2}}+\frac {2 (b B d-2 A b e+a B e)}{3 e (b d-a e)^2 (a+b x)^{3/2} \sqrt {d+e x}}-\frac {8 (b B d-2 A b e+a B e)}{3 (b d-a e)^3 \sqrt {a+b x} \sqrt {d+e x}}-\frac {16 e (b B d-2 A b e+a B e) \sqrt {a+b x}}{3 (b d-a e)^4 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.07, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37}
\begin {gather*} -\frac {2 (B d-A e)}{3 e (a+b x)^{3/2} (d+e x)^{3/2} (b d-a e)}-\frac {16 e \sqrt {a+b x} (a B e-2 A b e+b B d)}{3 \sqrt {d+e x} (b d-a e)^4}-\frac {8 (a B e-2 A b e+b B d)}{3 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3}+\frac {2 (a B e-2 A b e+b B d)}{3 e (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e)}{3 e (b d-a e) (a+b x)^{3/2} (d+e x)^{3/2}}-\frac {(b B d-2 A b e+a B e) \int \frac {1}{(a+b x)^{5/2} (d+e x)^{3/2}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (a+b x)^{3/2} (d+e x)^{3/2}}+\frac {2 (b B d-2 A b e+a B e)}{3 e (b d-a e)^2 (a+b x)^{3/2} \sqrt {d+e x}}+\frac {(4 (b B d-2 A b e+a B e)) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx}{3 (b d-a e)^2}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (a+b x)^{3/2} (d+e x)^{3/2}}+\frac {2 (b B d-2 A b e+a B e)}{3 e (b d-a e)^2 (a+b x)^{3/2} \sqrt {d+e x}}-\frac {8 (b B d-2 A b e+a B e)}{3 (b d-a e)^3 \sqrt {a+b x} \sqrt {d+e x}}-\frac {(8 e (b B d-2 A b e+a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{3 (b d-a e)^3}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (a+b x)^{3/2} (d+e x)^{3/2}}+\frac {2 (b B d-2 A b e+a B e)}{3 e (b d-a e)^2 (a+b x)^{3/2} \sqrt {d+e x}}-\frac {8 (b B d-2 A b e+a B e)}{3 (b d-a e)^3 \sqrt {a+b x} \sqrt {d+e x}}-\frac {16 e (b B d-2 A b e+a B e) \sqrt {a+b x}}{3 (b d-a e)^4 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 197, normalized size = 1.06 \begin {gather*} -\frac {2 \left (-B d e^2 (a+b x)^3+A e^3 (a+b x)^3+6 b B d e (a+b x)^2 (d+e x)-9 A b e^2 (a+b x)^2 (d+e x)+3 a B e^2 (a+b x)^2 (d+e x)+3 b^2 B d (a+b x) (d+e x)^2-9 A b^2 e (a+b x) (d+e x)^2+6 a b B e (a+b x) (d+e x)^2+A b^3 (d+e x)^3-a b^2 B (d+e x)^3\right )}{3 (b d-a e)^4 (a+b x)^{3/2} (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 279, normalized size = 1.50
method | result | size |
default | \(-\frac {2 \left (-16 A \,b^{3} e^{3} x^{3}+8 B a \,b^{2} e^{3} x^{3}+8 B \,b^{3} d \,e^{2} x^{3}-24 A a \,b^{2} e^{3} x^{2}-24 A \,b^{3} d \,e^{2} x^{2}+12 B \,a^{2} b \,e^{3} x^{2}+24 B a \,b^{2} d \,e^{2} x^{2}+12 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x -36 A a \,b^{2} d \,e^{2} x -6 A \,b^{3} d^{2} e x +3 B \,a^{3} e^{3} x +21 B \,a^{2} b d \,e^{2} x +21 B a \,b^{2} d^{2} e x +3 B \,b^{3} d^{3} x +a^{3} A \,e^{3}-9 A \,a^{2} b d \,e^{2}-9 A a \,b^{2} d^{2} e +A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}+12 B \,a^{2} b \,d^{2} e +2 B a \,b^{2} d^{3}\right )}{3 \left (a e -b d \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {3}{2}}}\) | \(279\) |
gosper | \(-\frac {2 \left (-16 A \,b^{3} e^{3} x^{3}+8 B a \,b^{2} e^{3} x^{3}+8 B \,b^{3} d \,e^{2} x^{3}-24 A a \,b^{2} e^{3} x^{2}-24 A \,b^{3} d \,e^{2} x^{2}+12 B \,a^{2} b \,e^{3} x^{2}+24 B a \,b^{2} d \,e^{2} x^{2}+12 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x -36 A a \,b^{2} d \,e^{2} x -6 A \,b^{3} d^{2} e x +3 B \,a^{3} e^{3} x +21 B \,a^{2} b d \,e^{2} x +21 B a \,b^{2} d^{2} e x +3 B \,b^{3} d^{3} x +a^{3} A \,e^{3}-9 A \,a^{2} b d \,e^{2}-9 A a \,b^{2} d^{2} e +A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}+12 B \,a^{2} b \,d^{2} e +2 B a \,b^{2} d^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {3}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(320\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 557 vs.
\(2 (176) = 352\).
time = 5.10, size = 557, normalized size = 2.99 \begin {gather*} -\frac {2 \, {\left (3 \, B b^{3} d^{3} x + {\left (2 \, B a b^{2} + A b^{3}\right )} d^{3} + {\left (A a^{3} + 8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x\right )} e^{3} + {\left (8 \, B b^{3} d x^{3} + 24 \, {\left (B a b^{2} - A b^{3}\right )} d x^{2} + 3 \, {\left (7 \, B a^{2} b - 12 \, A a b^{2}\right )} d x + {\left (2 \, B a^{3} - 9 \, A a^{2} b\right )} d\right )} e^{2} + 3 \, {\left (4 \, B b^{3} d^{2} x^{2} + {\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} d^{2} x + {\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{3 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}} \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 {A + B x}{\left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1102 vs.
\(2 (176) = 352\).
time = 0.96, size = 1102, normalized size = 5.92 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, B b^{7} d^{4} {\left | b \right |} e^{3} - 12 \, B a b^{6} d^{3} {\left | b \right |} e^{4} - 8 \, A b^{7} d^{3} {\left | b \right |} e^{4} + 6 \, B a^{2} b^{5} d^{2} {\left | b \right |} e^{5} + 24 \, A a b^{6} d^{2} {\left | b \right |} e^{5} + 4 \, B a^{3} b^{4} d {\left | b \right |} e^{6} - 24 \, A a^{2} b^{5} d {\left | b \right |} e^{6} - 3 \, B a^{4} b^{3} {\left | b \right |} e^{7} + 8 \, A a^{3} b^{4} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{9} d^{7} e - 7 \, a b^{8} d^{6} e^{2} + 21 \, a^{2} b^{7} d^{5} e^{3} - 35 \, a^{3} b^{6} d^{4} e^{4} + 35 \, a^{4} b^{5} d^{3} e^{5} - 21 \, a^{5} b^{4} d^{2} e^{6} + 7 \, a^{6} b^{3} d e^{7} - a^{7} b^{2} e^{8}} + \frac {3 \, {\left (2 \, B b^{8} d^{5} {\left | b \right |} e^{2} - 7 \, B a b^{7} d^{4} {\left | b \right |} e^{3} - 3 \, A b^{8} d^{4} {\left | b \right |} e^{3} + 8 \, B a^{2} b^{6} d^{3} {\left | b \right |} e^{4} + 12 \, A a b^{7} d^{3} {\left | b \right |} e^{4} - 2 \, B a^{3} b^{5} d^{2} {\left | b \right |} e^{5} - 18 \, A a^{2} b^{6} d^{2} {\left | b \right |} e^{5} - 2 \, B a^{4} b^{4} d {\left | b \right |} e^{6} + 12 \, A a^{3} b^{5} d {\left | b \right |} e^{6} + B a^{5} b^{3} {\left | b \right |} e^{7} - 3 \, A a^{4} b^{4} {\left | b \right |} e^{7}\right )}}{b^{9} d^{7} e - 7 \, a b^{8} d^{6} e^{2} + 21 \, a^{2} b^{7} d^{5} e^{3} - 35 \, a^{3} b^{6} d^{4} e^{4} + 35 \, a^{4} b^{5} d^{3} e^{5} - 21 \, a^{5} b^{4} d^{2} e^{6} + 7 \, a^{6} b^{3} d e^{7} - a^{7} b^{2} e^{8}}\right )}}{3 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (3 \, B b^{\frac {15}{2}} d^{3} e^{\frac {1}{2}} - B a b^{\frac {13}{2}} d^{2} e^{\frac {3}{2}} - 8 \, A b^{\frac {15}{2}} d^{2} e^{\frac {3}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac {11}{2}} d^{2} e^{\frac {1}{2}} - 7 \, B a^{2} b^{\frac {11}{2}} d e^{\frac {5}{2}} + 16 \, A a b^{\frac {13}{2}} d e^{\frac {5}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b^{\frac {9}{2}} d e^{\frac {3}{2}} + 18 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac {11}{2}} d e^{\frac {3}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{\frac {7}{2}} d e^{\frac {1}{2}} + 5 \, B a^{3} b^{\frac {9}{2}} e^{\frac {7}{2}} - 8 \, A a^{2} b^{\frac {11}{2}} e^{\frac {7}{2}} + 12 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{\frac {7}{2}} e^{\frac {5}{2}} - 18 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A a b^{\frac {9}{2}} e^{\frac {5}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B a b^{\frac {5}{2}} e^{\frac {3}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{\frac {7}{2}} e^{\frac {3}{2}}\right )}}{3 \, {\left (b^{3} d^{3} {\left | b \right |} - 3 \, a b^{2} d^{2} {\left | b \right |} e + 3 \, a^{2} b d {\left | b \right |} e^{2} - a^{3} {\left | b \right |} e^{3}\right )} {\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.50, size = 304, normalized size = 1.63 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {16\,b\,x^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^4}+\frac {4\,B\,a^3\,d\,e^2+2\,A\,a^3\,e^3+24\,B\,a^2\,b\,d^2\,e-18\,A\,a^2\,b\,d\,e^2+4\,B\,a\,b^2\,d^3-18\,A\,a\,b^2\,d^2\,e+2\,A\,b^3\,d^3}{3\,b\,e^2\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,x^2\,\left (a\,e+b\,d\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{e\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,x\,\left (a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{b\,e^2\,{\left (a\,e-b\,d\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,d^2\,\sqrt {a+b\,x}}{b\,e^2}+\frac {x^2\,\left (a\,e+2\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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