Optimal. Leaf size=225 \[ -\frac{16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.183504, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {777, 614, 613} \[ -\frac{16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 777
Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{\left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{\left (8 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.69248, size = 200, normalized size = 0.89 \[ \frac{2 \left (-3 \left (b^2-4 a c\right )^2 (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))+\left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x)) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )-8 c (b+2 c x) (a+x (b+c x))^2 \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )\right )}{15 c \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 608, normalized size = 2.7 \begin{align*} -{\frac{256\,Ab{c}^{4}e{x}^{5}-512\,A{c}^{5}d{x}^{5}-128\,Ba{c}^{4}e{x}^{5}-96\,B{b}^{2}{c}^{3}e{x}^{5}+256\,Bb{c}^{4}d{x}^{5}+640\,A{b}^{2}{c}^{3}e{x}^{4}-1280\,Ab{c}^{4}d{x}^{4}-320\,Bab{c}^{3}e{x}^{4}-240\,B{b}^{3}{c}^{2}e{x}^{4}+640\,B{b}^{2}{c}^{3}d{x}^{4}+640\,Aab{c}^{3}e{x}^{3}-1280\,Aa{c}^{4}d{x}^{3}+480\,A{b}^{3}{c}^{2}e{x}^{3}-960\,A{b}^{2}{c}^{3}d{x}^{3}-320\,B{a}^{2}{c}^{3}e{x}^{3}-480\,Ba{b}^{2}{c}^{2}e{x}^{3}+640\,Bab{c}^{3}d{x}^{3}-180\,B{b}^{4}ce{x}^{3}+480\,B{b}^{3}{c}^{2}d{x}^{3}+960\,Aa{b}^{2}{c}^{2}e{x}^{2}-1920\,Aab{c}^{3}d{x}^{2}+80\,A{b}^{4}ce{x}^{2}-160\,A{b}^{3}{c}^{2}d{x}^{2}-480\,B{a}^{2}b{c}^{2}e{x}^{2}-400\,Ba{b}^{3}ce{x}^{2}+960\,Ba{b}^{2}{c}^{2}d{x}^{2}-30\,B{b}^{5}e{x}^{2}+80\,B{b}^{4}cd{x}^{2}+480\,A{a}^{2}b{c}^{2}ex-960\,A{a}^{2}{c}^{3}dx+240\,Aa{b}^{3}cex-480\,Aa{b}^{2}{c}^{2}dx-10\,A{b}^{5}ex+20\,A{b}^{4}cdx-480\,B{a}^{2}{b}^{2}cex+480\,B{a}^{2}b{c}^{2}dx-40\,Ba{b}^{4}ex+240\,Ba{b}^{3}cdx-10\,B{b}^{5}dx+192\,A{a}^{3}{c}^{2}e+96\,A{a}^{2}{b}^{2}ce-480\,A{a}^{2}b{c}^{2}d-4\,Aa{b}^{4}e+80\,Aa{b}^{3}cd-6\,A{b}^{5}d-192\,B{a}^{3}bce+192\,B{a}^{3}{c}^{2}d-16\,B{a}^{2}{b}^{3}e+96\,B{a}^{2}{b}^{2}cd-4\,Ba{b}^{4}d}{960\,{a}^{3}{c}^{3}-720\,{a}^{2}{b}^{2}{c}^{2}+180\,a{b}^{4}c-15\,{b}^{6}} \left ( c{x}^{2}+bx+a \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 [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 [B] time = 1.18139, size = 1030, normalized size = 4.58 \begin{align*} \frac{{\left ({\left (2 \,{\left (4 \,{\left (\frac{2 \,{\left (8 \, B b c^{4} d - 16 \, A c^{5} d - 3 \, B b^{2} c^{3} e - 4 \, B a c^{4} e + 8 \, A b c^{4} e\right )} x}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}} + \frac{5 \,{\left (8 \, B b^{2} c^{3} d - 16 \, A b c^{4} d - 3 \, B b^{3} c^{2} e - 4 \, B a b c^{3} e + 8 \, A b^{2} c^{3} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x + \frac{5 \,{\left (24 \, B b^{3} c^{2} d + 32 \, B a b c^{3} d - 48 \, A b^{2} c^{3} d - 64 \, A a c^{4} d - 9 \, B b^{4} c e - 24 \, B a b^{2} c^{2} e + 24 \, A b^{3} c^{2} e - 16 \, B a^{2} c^{3} e + 32 \, A a b c^{3} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x + \frac{5 \,{\left (8 \, B b^{4} c d + 96 \, B a b^{2} c^{2} d - 16 \, A b^{3} c^{2} d - 192 \, A a b c^{3} d - 3 \, B b^{5} e - 40 \, B a b^{3} c e + 8 \, A b^{4} c e - 48 \, B a^{2} b c^{2} e + 96 \, A a b^{2} c^{2} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x - \frac{5 \,{\left (B b^{5} d - 24 \, B a b^{3} c d - 2 \, A b^{4} c d - 48 \, B a^{2} b c^{2} d + 48 \, A a b^{2} c^{2} d + 96 \, A a^{2} c^{3} d + 4 \, B a b^{4} e + A b^{5} e + 48 \, B a^{2} b^{2} c e - 24 \, A a b^{3} c e - 48 \, A a^{2} b c^{2} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x - \frac{2 \, B a b^{4} d + 3 \, A b^{5} d - 48 \, B a^{2} b^{2} c d - 40 \, A a b^{3} c d - 96 \, B a^{3} c^{2} d + 240 \, A a^{2} b c^{2} d + 8 \, B a^{2} b^{3} e + 2 \, A a b^{4} e + 96 \, B a^{3} b c e - 48 \, A a^{2} b^{2} c e - 96 \, A a^{3} c^{2} e}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}}{15 \,{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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