Optimal. Leaf size=145 \[ -\frac {2 (B d-A e) \sqrt {a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {4 b (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^3 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 \sqrt {a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac {4 b \sqrt {a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt {d+e x} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (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}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx &=-\frac {2 (B d-A e) \sqrt {a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {(b B d+4 A b e-5 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {(2 b (b B d+4 A b e-5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {4 b (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 134, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {a+b x} \left (15 A b^2-15 a b B-\frac {3 B d e (a+b x)^2}{(d+e x)^2}+\frac {3 A e^2 (a+b x)^2}{(d+e x)^2}+\frac {5 b B d (a+b x)}{d+e x}-\frac {10 A b e (a+b x)}{d+e x}+\frac {5 a B e (a+b x)}{d+e x}\right )}{15 (b d-a e)^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 149, normalized size = 1.03
method | result | size |
default | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} e^{2} x^{2}-10 B a b \,e^{2} x^{2}+2 B \,b^{2} d e \,x^{2}-4 A a b \,e^{2} x +20 A \,b^{2} d e x +5 B \,a^{2} e^{2} x -26 B a b d e x +5 B \,b^{2} d^{2} x +3 a^{2} A \,e^{2}-10 A a b d e +15 A \,b^{2} d^{2}+2 B \,a^{2} d e -10 B a b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{3}}\) | \(149\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} e^{2} x^{2}-10 B a b \,e^{2} x^{2}+2 B \,b^{2} d e \,x^{2}-4 A a b \,e^{2} x +20 A \,b^{2} d e x +5 B \,a^{2} e^{2} x -26 B a b d e x +5 B \,b^{2} d^{2} x +3 a^{2} A \,e^{2}-10 A a b d e +15 A \,b^{2} d^{2}+2 B \,a^{2} d e -10 B a b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(177\) |
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. 307 vs.
\(2 (135) = 270\).
time = 2.88, size = 307, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left (5 \, B b^{2} d^{2} x - 5 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} d^{2} + {\left (3 \, A a^{2} - 2 \, {\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} e^{2} + 2 \, {\left (B b^{2} d x^{2} - {\left (13 \, B a b - 10 \, A b^{2}\right )} d x + {\left (B a^{2} - 5 \, A a b\right )} d\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15 \, {\left (b^{3} d^{6} - a^{3} x^{3} e^{6} + 3 \, {\left (a^{2} b d x^{3} - a^{3} d x^{2}\right )} e^{5} - 3 \, {\left (a b^{2} d^{2} x^{3} - 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} e^{4} + {\left (b^{3} d^{3} x^{3} - 9 \, a b^{2} d^{3} x^{2} + 9 \, a^{2} b d^{3} x - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (b^{3} d^{4} x^{2} - 3 \, a b^{2} d^{4} x + a^{2} b d^{4}\right )} e^{2} + 3 \, {\left (b^{3} d^{5} x - a b^{2} 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}{\sqrt {a + b x} \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. 353 vs.
\(2 (135) = 270\).
time = 0.86, size = 353, normalized size = 2.43 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{6} d {\left | b \right |} e^{3} - 5 \, B a b^{5} {\left | b \right |} e^{4} + 4 \, A b^{6} {\left | b \right |} e^{4}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}} + \frac {5 \, {\left (B b^{7} d^{2} {\left | b \right |} e^{2} - 6 \, B a b^{6} d {\left | b \right |} e^{3} + 4 \, A b^{7} d {\left | b \right |} e^{3} + 5 \, B a^{2} b^{5} {\left | b \right |} e^{4} - 4 \, A a b^{6} {\left | b \right |} e^{4}\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}}\right )} - \frac {15 \, {\left (B a b^{7} d^{2} {\left | b \right |} e^{2} - A b^{8} d^{2} {\left | b \right |} e^{2} - 2 \, B a^{2} b^{6} d {\left | b \right |} e^{3} + 2 \, A a b^{7} d {\left | b \right |} e^{3} + B a^{3} b^{5} {\left | b \right |} e^{4} - A a^{2} b^{6} {\left | b \right |} e^{4}\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}}\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 = 2.25, size = 283, normalized size = 1.95 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^3\,d\,e+6\,A\,a^3\,e^2-20\,B\,a^2\,b\,d^2-20\,A\,a^2\,b\,d\,e+30\,A\,a\,b^2\,d^2}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\left (10\,B\,a^3\,e^2-48\,B\,a^2\,b\,d\,e-2\,A\,a^2\,b\,e^2-10\,B\,a\,b^2\,d^2+20\,A\,a\,b^2\,d\,e+30\,A\,b^3\,d^2\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^2\,x^3\,\left (4\,A\,b\,e-5\,B\,a\,e+B\,b\,d\right )}{15\,e^2\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,b\,x^2\,\left (a\,e+5\,b\,d\right )\,\left (4\,A\,b\,e-5\,B\,a\,e+B\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}\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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