Optimal. Leaf size=162 \[ \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^{5/2}}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}} \]
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Rubi [A] time = 0.08, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^{5/2}}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{(d+e x)^{7/2}} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e)}{e^2 (d+e x)^{7/2}}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^{5/2}}+\frac {b^2 B}{e^2 (d+e x)^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^{5/2}}+\frac {2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.53 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (a e (3 A e+2 B d+5 B e x)+A b e (2 d+5 e x)+b B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (a+b x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 40.00, size = 112, normalized size = 0.69 \begin {gather*} -\frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3 a A e^2+5 a B e (d+e x)-3 a B d e+5 A b e (d+e x)-3 A b d e+3 b B d^2-10 b B d (d+e x)+15 b B (d+e x)^2\right )}{15 e^2 (d+e x)^{5/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 101, normalized size = 0.62 \begin {gather*} -\frac {2 \, {\left (15 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 3 \, A a e^{2} + 2 \, {\left (B a + A b\right )} d e + 5 \, {\left (4 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 135, normalized size = 0.83 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b \mathrm {sgn}\left (b x + a\right ) - 10 \, {\left (x e + d\right )} B b d \mathrm {sgn}\left (b x + a\right ) + 3 \, B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (x e + d\right )} B a e \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (x e + d\right )} A b e \mathrm {sgn}\left (b x + a\right ) - 3 \, B a d e \mathrm {sgn}\left (b x + a\right ) - 3 \, A b d e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 0.55 \begin {gather*} -\frac {2 \left (15 B b \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x +20 B b d e x +3 A a \,e^{2}+2 A b d e +2 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right ) e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 118, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} A}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )} \sqrt {e x + d}} - \frac {2 \, {\left (15 \, b e^{2} x^{2} + 8 \, b d^{2} + 2 \, a d e + 5 \, {\left (4 \, b d e + a e^{2}\right )} x\right )} B}{15 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.52, size = 174, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{e^3}+\frac {6\,A\,a\,e^2+16\,B\,b\,d^2+4\,A\,b\,d\,e+4\,B\,a\,d\,e}{15\,b\,e^5}+\frac {x\,\left (10\,A\,b\,e^2+10\,B\,a\,e^2+40\,B\,b\,d\,e\right )}{15\,b\,e^5}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (15\,a\,e^5+30\,b\,d\,e^4\right )\,\sqrt {d+e\,x}}{15\,b\,e^5}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \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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