Optimal. Leaf size=246 \[ -\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac {32 b e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt {d+e x} (b d-a e)^5}-\frac {16 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac {4 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac {2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 246, 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 = {78, 45, 37} \begin {gather*} -\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}-\frac {32 b e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 \sqrt {d+e x} (b d-a e)^5}-\frac {16 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{15 (d+e x)^{3/2} (b d-a e)^4}-\frac {4 e \sqrt {a+b x} (5 a B e-8 A b e+3 b B d)}{5 b (d+e x)^{5/2} (b d-a e)^3}-\frac {2 (5 a B e-8 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}+\frac {(3 b B d-8 A b e+5 a B e) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {(2 e (3 b B d-8 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)^2}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {(8 e (3 b B d-8 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)^3}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {(16 b e (3 b B d-8 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)^4}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {32 b e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^5 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 134, normalized size = 0.54 \begin {gather*} \frac {2 \left ((a+b x) \left (2 e (a+b x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right ) (-5 a B e+8 A b e-3 b B d)-5 (A b-a B) (b d-a e)^4\right )}{15 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 276, normalized size = 1.12 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-\frac {5 A b^4 (d+e x)^4}{(a+b x)^4}+\frac {60 A b^3 e (d+e x)^3}{(a+b x)^3}+\frac {90 A b^2 e^2 (d+e x)^2}{(a+b x)^2}-\frac {20 A b e^3 (d+e x)}{a+b x}+\frac {5 a b^3 B (d+e x)^4}{(a+b x)^4}-\frac {15 b^3 B d (d+e x)^3}{(a+b x)^3}-\frac {45 a b^2 B e (d+e x)^3}{(a+b x)^3}-\frac {45 b^2 B d e (d+e x)^2}{(a+b x)^2}+\frac {5 a B e^3 (d+e x)}{a+b x}-\frac {45 a b B e^2 (d+e x)^2}{(a+b x)^2}+\frac {15 b B d e^2 (d+e x)}{a+b x}+3 A e^4-3 B d e^3\right )}{15 (d+e x)^{5/2} (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 150.35, size = 916, normalized size = 3.72 \begin {gather*} \frac {2 \, {\left (3 \, A a^{4} e^{4} - 5 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{4} - 30 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{3} e - 30 \, {\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{4} - 10 \, A a^{3} b\right )} d e^{3} - 16 \, {\left (3 \, B b^{4} d e^{3} + {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{4} - 8 \, {\left (15 \, B b^{4} d^{2} e^{2} + 2 \, {\left (17 \, B a b^{3} - 20 \, A b^{4}\right )} d e^{3} + 3 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x^{3} - 6 \, {\left (15 \, B b^{4} d^{3} e + 5 \, {\left (11 \, B a b^{3} - 8 \, A b^{4}\right )} d^{2} e^{2} + {\left (53 \, B a^{2} b^{2} - 80 \, A a b^{3}\right )} d e^{3} + {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} - {\left (15 \, B b^{4} d^{4} + 40 \, {\left (4 \, B a b^{3} - A b^{4}\right )} d^{3} e + 90 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 24 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (a^{2} b^{5} d^{8} - 5 \, a^{3} b^{4} d^{7} e + 10 \, a^{4} b^{3} d^{6} e^{2} - 10 \, a^{5} b^{2} d^{5} e^{3} + 5 \, a^{6} b d^{4} e^{4} - a^{7} d^{3} e^{5} + {\left (b^{7} d^{5} e^{3} - 5 \, a b^{6} d^{4} e^{4} + 10 \, a^{2} b^{5} d^{3} e^{5} - 10 \, a^{3} b^{4} d^{2} e^{6} + 5 \, a^{4} b^{3} d e^{7} - a^{5} b^{2} e^{8}\right )} x^{5} + {\left (3 \, b^{7} d^{6} e^{2} - 13 \, a b^{6} d^{5} e^{3} + 20 \, a^{2} b^{5} d^{4} e^{4} - 10 \, a^{3} b^{4} d^{3} e^{5} - 5 \, a^{4} b^{3} d^{2} e^{6} + 7 \, a^{5} b^{2} d e^{7} - 2 \, a^{6} b e^{8}\right )} x^{4} + {\left (3 \, b^{7} d^{7} e - 9 \, a b^{6} d^{6} e^{2} + a^{2} b^{5} d^{5} e^{3} + 25 \, a^{3} b^{4} d^{4} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{5} + 17 \, a^{5} b^{2} d^{2} e^{6} - a^{6} b d e^{7} - a^{7} e^{8}\right )} x^{3} + {\left (b^{7} d^{8} + a b^{6} d^{7} e - 17 \, a^{2} b^{5} d^{6} e^{2} + 35 \, a^{3} b^{4} d^{5} e^{3} - 25 \, a^{4} b^{3} d^{4} e^{4} - a^{5} b^{2} d^{3} e^{5} + 9 \, a^{6} b d^{2} e^{6} - 3 \, a^{7} d e^{7}\right )} x^{2} + {\left (2 \, a b^{6} d^{8} - 7 \, a^{2} b^{5} d^{7} e + 5 \, a^{3} b^{4} d^{6} e^{2} + 10 \, a^{4} b^{3} d^{5} e^{3} - 20 \, a^{5} b^{2} d^{4} e^{4} + 13 \, a^{6} b d^{3} e^{5} - 3 \, a^{7} d^{2} e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 9.05, size = 1936, normalized size = 7.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 505, normalized size = 2.05 \begin {gather*} -\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {5}{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 )} \end {gather*}
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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mupad [B] time = 2.81, size = 444, normalized size = 1.80 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {32\,b^2\,x^4\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,{\left (a\,e-b\,d\right )}^5}+\frac {-4\,B\,a^4\,d\,e^3-6\,A\,a^4\,e^4+60\,B\,a^3\,b\,d^2\,e^2+40\,A\,a^3\,b\,d\,e^3+180\,B\,a^2\,b^2\,d^3\,e-180\,A\,a^2\,b^2\,d^2\,e^2+20\,B\,a\,b^3\,d^4-120\,A\,a\,b^3\,d^3\,e+10\,A\,b^4\,d^4}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,x^2\,\left (a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{5\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+15\,a^2\,b\,d\,e^2+45\,a\,b^2\,d^2\,e+5\,b^3\,d^3\right )}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b\,x^3\,\left (3\,a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a\,d^3\,\sqrt {a+b\,x}}{b\,e^3}+\frac {x^3\,\left (a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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