Optimal. Leaf size=187 \[ -\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.07, antiderivative size = 187, 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 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}+\frac {16 b \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt {d+e x} (b d-a e)^4}+\frac {8 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (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)^{3/2} (d+e x)^{7/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {(b B d-6 A b e+5 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {(4 (b B d-6 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {(8 b (b B d-6 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 199, normalized size = 1.06 \begin {gather*} -\frac {2 \left (-3 B d e^2 (a+b x)^3+3 A e^3 (a+b x)^3+10 b B d e (a+b x)^2 (d+e x)-15 A b e^2 (a+b x)^2 (d+e x)+5 a B e^2 (a+b x)^2 (d+e x)-15 b^2 B d (a+b x) (d+e x)^2+45 A b^2 e (a+b x) (d+e x)^2-30 a b B e (a+b x) (d+e x)^2+15 A b^3 (d+e x)^3-15 a b^2 B (d+e x)^3\right )}{15 (b d-a e)^4 \sqrt {a+b x} (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 281, normalized size = 1.50
method | result | size |
default | \(-\frac {2 \left (48 A \,b^{3} e^{3} x^{3}-40 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}+120 A \,b^{3} d \,e^{2} x^{2}-20 B \,a^{2} b \,e^{3} x^{2}-104 B a \,b^{2} d \,e^{2} x^{2}-20 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x +60 A a \,b^{2} d \,e^{2} x +90 A \,b^{3} d^{2} e x +5 B \,a^{3} e^{3} x -49 B \,a^{2} b d \,e^{2} x -85 B a \,b^{2} d^{2} e x -15 B \,b^{3} d^{3} x +3 a^{3} A \,e^{3}-15 A \,a^{2} b d \,e^{2}+45 A a \,b^{2} d^{2} e +15 A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}-20 B \,a^{2} b \,d^{2} e -30 B a \,b^{2} d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \sqrt {b x +a}\, \left (a e -b d \right )^{4}}\) | \(281\) |
gosper | \(-\frac {2 \left (48 A \,b^{3} e^{3} x^{3}-40 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}+120 A \,b^{3} d \,e^{2} x^{2}-20 B \,a^{2} b \,e^{3} x^{2}-104 B a \,b^{2} d \,e^{2} x^{2}-20 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x +60 A a \,b^{2} d \,e^{2} x +90 A \,b^{3} d^{2} e x +5 B \,a^{3} e^{3} x -49 B \,a^{2} b d \,e^{2} x -85 B a \,b^{2} d^{2} e x -15 B \,b^{3} d^{3} x +3 a^{3} A \,e^{3}-15 A \,a^{2} b d \,e^{2}+45 A a \,b^{2} d^{2} e +15 A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}-20 B \,a^{2} b \,d^{2} e -30 B a \,b^{2} d^{3}\right )}{15 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {5}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
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. 573 vs.
\(2 (179) = 358\).
time = 5.94, size = 573, normalized size = 3.06 \begin {gather*} \frac {2 \, {\left (15 \, B b^{3} d^{3} x + 15 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{3} - {\left (3 \, A a^{3} - 8 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} e^{3} + {\left (8 \, B b^{3} d x^{3} + 8 \, {\left (13 \, B a b^{2} - 15 \, A b^{3}\right )} d x^{2} + {\left (49 \, B a^{2} b - 60 \, A a b^{2}\right )} d x - {\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} d\right )} e^{2} + 5 \, {\left (4 \, B b^{3} d^{2} x^{2} + {\left (17 \, B a b^{2} - 18 \, A b^{3}\right )} d^{2} x + {\left (4 \, B a^{2} b - 9 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\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 {3}{2}} \left (d + e x\right )^{\frac {7}{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. 1266 vs.
\(2 (179) = 358\).
time = 0.74, size = 1266, normalized size = 6.77 \begin {gather*} \frac {4 \, {\left (B^{2} a^{2} b^{7} e - 2 \, A B a b^{8} e + A^{2} b^{9} e\right )}}{{\left (B a b^{\frac {11}{2}} d e^{\frac {1}{2}} - A b^{\frac {13}{2}} d e^{\frac {1}{2}} - B a^{2} b^{\frac {9}{2}} e^{\frac {3}{2}} + A a b^{\frac {11}{2}} e^{\frac {3}{2}} - {\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 {7}{2}} e^{\frac {1}{2}} + {\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 {9}{2}} e^{\frac {1}{2}}\right )} {\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 )}} + \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {{\left (8 \, B b^{13} d^{6} e^{4} - 15 \, B a b^{12} d^{5} e^{5} - 33 \, A b^{13} d^{5} e^{5} - 45 \, B a^{2} b^{11} d^{4} e^{6} + 165 \, A a b^{12} d^{4} e^{6} + 170 \, B a^{3} b^{10} d^{3} e^{7} - 330 \, A a^{2} b^{11} d^{3} e^{7} - 210 \, B a^{4} b^{9} d^{2} e^{8} + 330 \, A a^{3} b^{10} d^{2} e^{8} + 117 \, B a^{5} b^{8} d e^{9} - 165 \, A a^{4} b^{9} d e^{9} - 25 \, B a^{6} b^{7} e^{10} + 33 \, A a^{5} b^{8} e^{10}\right )} {\left (b x + a\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}} + \frac {5 \, {\left (4 \, B b^{14} d^{7} e^{3} - 13 \, B a b^{13} d^{6} e^{4} - 15 \, A b^{14} d^{6} e^{4} - 6 \, B a^{2} b^{12} d^{5} e^{5} + 90 \, A a b^{13} d^{5} e^{5} + 85 \, B a^{3} b^{11} d^{4} e^{6} - 225 \, A a^{2} b^{12} d^{4} e^{6} - 160 \, B a^{4} b^{10} d^{3} e^{7} + 300 \, A a^{3} b^{11} d^{3} e^{7} + 141 \, B a^{5} b^{9} d^{2} e^{8} - 225 \, A a^{4} b^{10} d^{2} e^{8} - 62 \, B a^{6} b^{8} d e^{9} + 90 \, A a^{5} b^{9} d e^{9} + 11 \, B a^{7} b^{7} e^{10} - 15 \, A a^{6} b^{8} e^{10}\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}}\right )} + \frac {15 \, {\left (B b^{15} d^{8} e^{2} - 5 \, B a b^{14} d^{7} e^{3} - 3 \, A b^{15} d^{7} e^{3} + 7 \, B a^{2} b^{13} d^{6} e^{4} + 21 \, A a b^{14} d^{6} e^{4} + 7 \, B a^{3} b^{12} d^{5} e^{5} - 63 \, A a^{2} b^{13} d^{5} e^{5} - 35 \, B a^{4} b^{11} d^{4} e^{6} + 105 \, A a^{3} b^{12} d^{4} e^{6} + 49 \, B a^{5} b^{10} d^{3} e^{7} - 105 \, A a^{4} b^{11} d^{3} e^{7} - 35 \, B a^{6} b^{9} d^{2} e^{8} + 63 \, A a^{5} b^{10} d^{2} e^{8} + 13 \, B a^{7} b^{8} d e^{9} - 21 \, A a^{6} b^{9} d e^{9} - 2 \, B a^{8} b^{7} e^{10} + 3 \, A a^{7} b^{8} e^{10}\right )}}{b^{11} d^{9} {\left | b \right |} e^{2} - 9 \, a b^{10} d^{8} {\left | b \right |} e^{3} + 36 \, a^{2} b^{9} d^{7} {\left | b \right |} e^{4} - 84 \, a^{3} b^{8} d^{6} {\left | b \right |} e^{5} + 126 \, a^{4} b^{7} d^{5} {\left | b \right |} e^{6} - 126 \, a^{5} b^{6} d^{4} {\left | b \right |} e^{7} + 84 \, a^{6} b^{5} d^{3} {\left | b \right |} e^{8} - 36 \, a^{7} b^{4} d^{2} {\left | b \right |} e^{9} + 9 \, a^{8} b^{3} d {\left | b \right |} e^{10} - a^{9} b^{2} {\left | b \right |} e^{11}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.46, size = 288, normalized size = 1.54 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {2\,x\,\left (-a^2\,e^2+10\,a\,b\,d\,e+15\,b^2\,d^2\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {4\,B\,a^3\,d\,e^2+6\,A\,a^3\,e^3-40\,B\,a^2\,b\,d^2\,e-30\,A\,a^2\,b\,d\,e^2-60\,B\,a\,b^2\,d^3+90\,A\,a\,b^2\,d^2\,e+30\,A\,b^3\,d^3}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^2\,x^3\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b\,x^2\,\left (a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^2\,{\left (a\,e-b\,d\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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