Optimal. Leaf size=267 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)} \]
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Rubi [A] time = 0.32, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{13/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx}{11 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{33 (c d f-a e g)^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx}{231 (c d f-a e g)^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 152, normalized size = 0.57 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (-105 a^3 e^3 g^3+35 a^2 c d e^2 g^2 (11 f+2 g x)-5 a c^2 d^2 e g \left (99 f^2+44 f g x+8 g^2 x^2\right )+c^3 d^3 \left (231 f^3+198 f^2 g x+88 f g^2 x^2+16 g^3 x^3\right )\right )}{1155 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 1420, normalized size = 5.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 260, normalized size = 0.97 \[ -\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+40 a \,c^{2} d^{2} e \,g^{3} x^{2}-88 c^{3} d^{3} f \,g^{2} x^{2}-70 a^{2} c d \,e^{2} g^{3} x +220 a \,c^{2} d^{2} e f \,g^{2} x -198 c^{3} d^{3} f^{2} g x +105 a^{3} e^{3} g^{3}-385 a^{2} c d \,e^{2} f \,g^{2}+495 a \,c^{2} d^{2} e \,f^{2} g -231 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (g x +f \right )^{\frac {11}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.83, size = 519, normalized size = 1.94 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {210\,a^5\,e^5\,g^3-770\,a^4\,c\,d\,e^4\,f\,g^2+990\,a^3\,c^2\,d^2\,e^3\,f^2\,g-462\,a^2\,c^3\,d^3\,e^2\,f^3}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^2\,\left (-10\,a^3\,c^2\,d^2\,e^3\,g^3+66\,a^2\,c^3\,d^3\,e^2\,f\,g^2-198\,a\,c^4\,d^4\,e\,f^2\,g+462\,c^5\,d^5\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^5\,d^5\,x^5}{1155\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^3\,d^3\,x^3\,\left (3\,a^2\,e^2\,g^2-22\,a\,c\,d\,e\,f\,g+99\,c^2\,d^2\,f^2\right )}{1155\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{1155\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,a\,c\,d\,e\,x\,\left (70\,a^3\,e^3\,g^3-275\,a^2\,c\,d\,e^2\,f\,g^2+396\,a\,c^2\,d^2\,e\,f^2\,g-231\,c^3\,d^3\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {5\,f\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {5\,f^4\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {10\,f^2\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {10\,f^3\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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