Optimal. Leaf size=464 \[ \frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 \sqrt{c} e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{4 e^4 (d+e x)}-\frac{15 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{32 e^6}-\frac{15 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^7}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2} \]
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Rubi [A] time = 0.845349, antiderivative size = 464, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {812, 814, 843, 621, 206, 724} \[ \frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 \sqrt{c} e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{4 e^4 (d+e x)}-\frac{15 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{32 e^6}-\frac{15 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^7}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}-\frac{5 \int \frac{\left (4 \left (3 b c d-b^2 e-2 a c e\right )+12 c (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx}{16 e^2}\\ &=-\frac{5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}+\frac{5 \int \frac{\left (-12 \left (7 b^2 c d e+4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )\right )+12 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{32 e^4}\\ &=-\frac{15 \left (64 c^3 d^3-7 b^3 e^3+4 b c e^2 (14 b d-5 a e)-16 c^2 d e (7 b d-2 a e)-2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{32 e^6}-\frac{5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}-\frac{5 \int \frac{6 c \left (7 b^4 d e^3+16 a c^2 d e \left (4 c d^2+3 a e^2\right )+8 b^2 c d e \left (14 c d^2+11 a e^2\right )-8 b^3 \left (7 c d^2 e^2+a e^4\right )-32 b c \left (2 c^2 d^4+5 a c d^2 e^2+a^2 e^4\right )\right )-6 c \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 c e^6}\\ &=-\frac{15 \left (64 c^3 d^3-7 b^3 e^3+4 b c e^2 (14 b d-5 a e)-16 c^2 d e (7 b d-2 a e)-2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{32 e^6}-\frac{5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}+\frac{\left (15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{64 e^7}-\frac{\left (5 \left (6 c d \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )+6 c e \left (7 b^4 d e^3+16 a c^2 d e \left (4 c d^2+3 a e^2\right )+8 b^2 c d e \left (14 c d^2+11 a e^2\right )-8 b^3 \left (7 c d^2 e^2+a e^4\right )-32 b c \left (2 c^2 d^4+5 a c d^2 e^2+a^2 e^4\right )\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 c e^7}\\ &=-\frac{15 \left (64 c^3 d^3-7 b^3 e^3+4 b c e^2 (14 b d-5 a e)-16 c^2 d e (7 b d-2 a e)-2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{32 e^6}-\frac{5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}+\frac{\left (15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{32 e^7}+\frac{\left (5 \left (6 c d \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )+6 c e \left (7 b^4 d e^3+16 a c^2 d e \left (4 c d^2+3 a e^2\right )+8 b^2 c d e \left (14 c d^2+11 a e^2\right )-8 b^3 \left (7 c d^2 e^2+a e^4\right )-32 b c \left (2 c^2 d^4+5 a c d^2 e^2+a^2 e^4\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{64 c e^7}\\ &=-\frac{15 \left (64 c^3 d^3-7 b^3 e^3+4 b c e^2 (14 b d-5 a e)-16 c^2 d e (7 b d-2 a e)-2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{32 e^6}-\frac{5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac{(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}+\frac{15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 \sqrt{c} e^7}-\frac{15 (2 c d-b e) \sqrt{c d^2-b d e+a e^2} \left (8 c^2 d^2-8 b c d e+b^2 e^2+4 a c e^2\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{8 e^7}\\ \end{align*}
Mathematica [A] time = 2.26659, size = 535, normalized size = 1.15 \[ \frac{-\frac{2 e \sqrt{a+x (b+c x)} \left (2 c e^2 \left (16 a^2 e^2 (d+2 e x)-2 a b e \left (145 d^2+234 d e x+65 e^2 x^2\right )+b^2 \left (655 d^2 e x+420 d^3+166 d e^2 x^2-37 e^3 x^3\right )\right )+b e^3 \left (16 a^2 e^2+8 a b e (5 d+9 e x)+b^2 \left (-\left (105 d^2+170 d e x+49 e^2 x^2\right )\right )\right )-8 c^2 e \left (a e \left (-155 d^2 e x-100 d^3-38 d e^2 x^2+9 e^3 x^3\right )+b \left (75 d^2 e^2 x^2+320 d^3 e x+210 d^4-18 d e^3 x^3+7 e^4 x^4\right )\right )+16 c^3 \left (20 d^3 e^2 x^2-5 d^2 e^3 x^3+90 d^4 e x+60 d^5+2 d e^4 x^4-e^5 x^5\right )\right )}{(d+e x)^2}+\frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{\sqrt{c}}+120 (2 c d-b e) \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{64 e^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 18705, normalized size = 40.3 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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