Optimal. Leaf size=256 \[ \frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rubi [A] time = 0.233234, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 50, 63, 208} \[ \frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx &=-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{(2 b B d+7 A b e-9 a B e) \int \frac{(d+e x)^{7/2}}{a+b x} \, dx}{2 b (b d-a e)}\\ &=\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{(2 b B d+7 A b e-9 a B e) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{((b d-a e) (2 b B d+7 A b e-9 a B e)) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^2 (2 b B d+7 A b e-9 a B e)\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^4}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^5}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}+\frac{\left ((b d-a e)^3 (2 b B d+7 A b e-9 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^5 e}\\ &=\frac{(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt{d+e x}}{b^5}+\frac{(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac{(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac{(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac{(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.541665, size = 193, normalized size = 0.75 \[ \frac{\frac{2 \left (-\frac{9 a B e}{2}+\frac{7 A b e}{2}+b B d\right ) \left (7 (b d-a e) \left (5 (b d-a e) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )+15 b^{7/2} (d+e x)^{7/2}\right )}{105 b^{9/2}}+\frac{(d+e x)^{9/2} (a B-A b)}{a+b x}}{b (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 915, normalized size = 3.6 \begin{align*} \text{result too large to display} \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 = 1.45688, size = 2191, normalized size = 8.56 \begin{align*} \text{result too large to display} \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 = 1.96398, size = 815, normalized size = 3.18 \begin{align*} \frac{{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} + \frac{\sqrt{x e + d} B a b^{3} d^{3} e - \sqrt{x e + d} A b^{4} d^{3} e - 3 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B a^{3} b d e^{3} - 3 \, \sqrt{x e + d} A a^{2} b^{2} d e^{3} - \sqrt{x e + d} B a^{4} e^{4} + \sqrt{x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{12} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d^{2} + 105 \, \sqrt{x e + d} B b^{12} d^{3} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{11} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{12} e - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} d e - 630 \, \sqrt{x e + d} B a b^{11} d^{2} e + 315 \, \sqrt{x e + d} A b^{12} d^{2} e + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{10} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{11} e^{2} + 945 \, \sqrt{x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt{x e + d} A a b^{11} d e^{2} - 420 \, \sqrt{x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt{x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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