Optimal. Leaf size=237 \[ -\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^5 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {32 b^2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt {d+e x} (b d-a e)^5}+\frac {16 b \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac {12 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)} \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)^{9/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {(b B d-8 A b e+7 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {(6 (b B d-8 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {(24 b (b B d-8 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {\left (16 b^2 (b B d-8 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^5 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 270, normalized size = 1.14 \begin {gather*} -\frac {2 \left (5 B d e^3 (a+b x)^4-5 A e^4 (a+b x)^4-21 b B d e^2 (a+b x)^3 (d+e x)+28 A b e^3 (a+b x)^3 (d+e x)-7 a B e^3 (a+b x)^3 (d+e x)+35 b^2 B d e (a+b x)^2 (d+e x)^2-70 A b^2 e^2 (a+b x)^2 (d+e x)^2+35 a b B e^2 (a+b x)^2 (d+e x)^2-35 b^3 B d (a+b x) (d+e x)^3+140 A b^3 e (a+b x) (d+e x)^3-105 a b^2 B e (a+b x) (d+e x)^3+35 A b^4 (d+e x)^4-35 a b^3 B (d+e x)^4\right )}{35 (b d-a e)^5 \sqrt {a+b x} (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs.
\(2(209)=418\).
time = 0.09, size = 449, normalized size = 1.89
method | result | size |
default | \(-\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} \sqrt {b x +a}\, \left (a e -b d \right )^{5}}\) | \(449\) |
gosper | \(-\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {7}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(505\) |
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. 865 vs.
\(2 (227) = 454\).
time = 23.84, size = 865, normalized size = 3.65 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} d^{4} x + 35 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{4} + {\left (5 \, A a^{4} + 16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} e^{4} + 2 \, {\left (8 \, B b^{4} d x^{4} + 8 \, {\left (25 \, B a b^{3} - 28 \, A b^{4}\right )} d x^{3} + {\left (97 \, B a^{2} b^{2} - 112 \, A a b^{3}\right )} d x^{2} - 4 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d x + {\left (B a^{4} - 14 \, A a^{3} b\right )} d\right )} e^{3} + 14 \, {\left (4 \, B b^{4} d^{2} x^{3} + {\left (37 \, B a b^{3} - 40 \, A b^{4}\right )} d^{2} x^{2} + {\left (17 \, B a^{2} b^{2} - 20 \, A a b^{3}\right )} d^{2} x - {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} d^{2}\right )} e^{2} + 70 \, {\left (B b^{4} d^{3} x^{2} + 4 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} x + {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{35 \, {\left (b^{6} d^{9} x + a b^{5} d^{9} - {\left (a^{5} b x^{5} + a^{6} x^{4}\right )} e^{9} + {\left (5 \, a^{4} b^{2} d x^{5} + a^{5} b d x^{4} - 4 \, a^{6} d x^{3}\right )} e^{8} - 2 \, {\left (5 \, a^{3} b^{3} d^{2} x^{5} - 5 \, a^{4} b^{2} d^{2} x^{4} - 7 \, a^{5} b d^{2} x^{3} + 3 \, a^{6} d^{2} x^{2}\right )} e^{7} + 2 \, {\left (5 \, a^{2} b^{4} d^{3} x^{5} - 15 \, a^{3} b^{3} d^{3} x^{4} - 5 \, a^{4} b^{2} d^{3} x^{3} + 13 \, a^{5} b d^{3} x^{2} - 2 \, a^{6} d^{3} x\right )} e^{6} - {\left (5 \, a b^{5} d^{4} x^{5} - 35 \, a^{2} b^{4} d^{4} x^{4} + 20 \, a^{3} b^{3} d^{4} x^{3} + 40 \, a^{4} b^{2} d^{4} x^{2} - 19 \, a^{5} b d^{4} x + a^{6} d^{4}\right )} e^{5} + {\left (b^{6} d^{5} x^{5} - 19 \, a b^{5} d^{5} x^{4} + 40 \, a^{2} b^{4} d^{5} x^{3} + 20 \, a^{3} b^{3} d^{5} x^{2} - 35 \, a^{4} b^{2} d^{5} x + 5 \, a^{5} b d^{5}\right )} e^{4} + 2 \, {\left (2 \, b^{6} d^{6} x^{4} - 13 \, a b^{5} d^{6} x^{3} + 5 \, a^{2} b^{4} d^{6} x^{2} + 15 \, a^{3} b^{3} d^{6} x - 5 \, a^{4} b^{2} d^{6}\right )} e^{3} + 2 \, {\left (3 \, b^{6} d^{7} x^{3} - 7 \, a b^{5} d^{7} x^{2} - 5 \, a^{2} b^{4} d^{7} x + 5 \, a^{3} b^{3} d^{7}\right )} e^{2} + {\left (4 \, b^{6} d^{8} x^{2} - a b^{5} d^{8} x - 5 \, a^{2} b^{4} d^{8}\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 {9}{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. 2477 vs.
\(2 (227) = 454\).
time = 1.54, size = 2477, normalized size = 10.45 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 409, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^4\,d\,e^3+10\,A\,a^4\,e^4-28\,B\,a^3\,b\,d^2\,e^2-56\,A\,a^3\,b\,d\,e^3+140\,B\,a^2\,b^2\,d^3\,e+140\,A\,a^2\,b^2\,d^2\,e^2+140\,B\,a\,b^3\,d^4-280\,A\,a\,b^3\,d^3\,e-70\,A\,b^4\,d^4}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^3\,x^4\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )\,\left (a^3\,e^3-7\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b^2\,x^3\,\left (a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b\,x^2\,\left (-a^2\,e^2+14\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^3\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {d^4\,\sqrt {a+b\,x}}{e^4}+\frac {6\,d^2\,x^2\,\sqrt {a+b\,x}}{e^2}+\frac {4\,d\,x^3\,\sqrt {a+b\,x}}{e}+\frac {4\,d^3\,x\,\sqrt {a+b\,x}}{e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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