Optimal. Leaf size=187 \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)}+\frac{16 b \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt{d+e x} (b d-a e)^4}+\frac{8 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \]
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Rubi [A] time = 0.113252, antiderivative size = 187, normalized size of antiderivative = 1., 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 = {78, 45, 37} \[ -\frac{2 (A b-a B)}{b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)}+\frac{16 b \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt{d+e x} (b d-a e)^4}+\frac{8 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
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*}
Mathematica [A] time = 0.187986, size = 114, normalized size = 0.61 \[ \frac{2 \left (-(a+b x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right ) (-5 a B e+6 A b e-b B d)-15 (A b-a B) (b d-a e)^3\right )}{15 b \sqrt{a+b x} (d+e x)^{5/2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 322, normalized size = 1.7 \begin{align*} -{\frac{96\,A{b}^{3}{e}^{3}{x}^{3}-80\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}+240\,A{b}^{3}d{e}^{2}{x}^{2}-40\,B{a}^{2}b{e}^{3}{x}^{2}-208\,Ba{b}^{2}d{e}^{2}{x}^{2}-40\,B{b}^{3}{d}^{2}e{x}^{2}-12\,A{a}^{2}b{e}^{3}x+120\,Aa{b}^{2}d{e}^{2}x+180\,A{b}^{3}{d}^{2}ex+10\,B{a}^{3}{e}^{3}x-98\,B{a}^{2}bd{e}^{2}x-170\,Ba{b}^{2}{d}^{2}ex-30\,B{b}^{3}{d}^{3}x+6\,A{a}^{3}{e}^{3}-30\,A{a}^{2}bd{e}^{2}+90\,Aa{b}^{2}{d}^{2}e+30\,A{b}^{3}{d}^{3}+4\,B{a}^{3}d{e}^{2}-40\,B{a}^{2}b{d}^{2}e-60\,Ba{b}^{2}{d}^{3}}{15\,{e}^{4}{a}^{4}-60\,b{e}^{3}d{a}^{3}+90\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-60\,a{b}^{3}{d}^{3}e+15\,{b}^{4}{d}^{4}}{\frac{1}{\sqrt{bx+a}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 51.4411, size = 1193, normalized size = 6.38 \begin{align*} -\frac{2 \,{\left (3 \, A a^{3} e^{3} - 15 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{3} - 5 \,{\left (4 \, B a^{2} b - 9 \, A a b^{2}\right )} d^{2} e +{\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} d e^{2} - 8 \,{\left (B b^{3} d e^{2} +{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \,{\left (5 \, B b^{3} d^{2} e + 2 \,{\left (13 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} +{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (15 \, B b^{3} d^{3} + 5 \,{\left (17 \, B a b^{2} - 18 \, A b^{3}\right )} d^{2} e +{\left (49 \, B a^{2} b - 60 \, A a b^{2}\right )} d e^{2} -{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.97222, size = 1183, normalized size = 6.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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