Optimal. Leaf size=574 \[ -\frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (48 b^2 c d e^2-b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (10 A c e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac{d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]
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Rubi [A] time = 0.927577, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {812, 814, 843, 620, 206, 724} \[ -\frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (48 b^2 c d e^2-b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (10 A c e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac{d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
Rule 843
Rule 620
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac{\int \frac{(b (6 B d-5 A e)+(12 B c d-b B e-10 A c e) x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e^2}\\ &=-\frac{\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac{\int \frac{\left (\frac{1}{2} b d \left (10 A c e (8 c d-7 b e)-2 B \left (48 c^2 d^2-46 b c d e+\frac{3 b^2 e^2}{2}\right )\right )+\frac{1}{2} \left (8 b c e (6 B d-5 A e) (2 c d-b e)-2 (12 B c d-b B e-10 A c e) \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}\right )\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{16 c e^4}\\ &=-\frac{\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^6}-\frac{\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac{\int \frac{-\frac{1}{4} b d \left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )\right )-\frac{1}{4} \left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{64 c^2 e^6}\\ &=-\frac{\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^6}-\frac{\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac{\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 e^7}+\frac{\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^2 e^7}\\ &=-\frac{\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^6}-\frac{\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac{\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{e^7}+\frac{\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{128 c^2 e^7}\\ &=-\frac{\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^6}-\frac{\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac{(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac{\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^7}+\frac{d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{2 e^7}\\ \end{align*}
Mathematica [B] time = 6.11654, size = 1932, normalized size = 3.37 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 7095, normalized size = 12.4 \begin{align*} \text{output 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 [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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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 [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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