Optimal. Leaf size=307 \[ \frac {x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac {a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac {a^2 x \sqrt {a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac {a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}+\frac {e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac {13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]
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Rubi [A] time = 0.29, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {743, 833, 780, 195, 217, 206} \begin {gather*} \frac {x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac {a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac {a^2 x \sqrt {a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac {a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}+\frac {e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac {13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 743
Rule 780
Rule 833
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {\int (d+e x)^2 \left (10 c d^2-3 a e^2+13 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {\int (d+e x) \left (c d \left (90 c d^2-53 a e^2\right )+c e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{96 c^2}\\ &=\frac {a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \sqrt {a+c x^2} \, dx}{128 c^2}\\ &=\frac {a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt {a+c x^2}}{256 c^2}+\frac {a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{256 c^2}\\ &=\frac {a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt {a+c x^2}}{256 c^2}+\frac {a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{256 c^2}\\ &=\frac {a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt {a+c x^2}}{256 c^2}+\frac {a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 283, normalized size = 0.92 \begin {gather*} \frac {a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{256 c^{5/2}}+\frac {\sqrt {a+c x^2} \left (-5 a^4 e^3 (2048 d+189 e x)+10 a^3 c e \left (4608 d^3+1890 d^2 e x+512 d e^2 x^2+63 e^3 x^3\right )+24 a^2 c^2 x \left (2310 d^4+5760 d^3 e x+6195 d^2 e^2 x^2+3200 d e^3 x^3+651 e^4 x^4\right )+16 a c^3 x^3 \left (2730 d^4+8640 d^3 e x+10710 d^2 e^2 x^2+6080 d e^3 x^3+1323 e^4 x^4\right )+64 c^4 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{80640 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.83, size = 346, normalized size = 1.13 \begin {gather*} \frac {\left (-3 a^5 e^4+60 a^4 c d^2 e^2-80 a^3 c^2 d^4\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{256 c^{5/2}}+\frac {\sqrt {a+c x^2} \left (-10240 a^4 d e^3-945 a^4 e^4 x+46080 a^3 c d^3 e+18900 a^3 c d^2 e^2 x+5120 a^3 c d e^3 x^2+630 a^3 c e^4 x^3+55440 a^2 c^2 d^4 x+138240 a^2 c^2 d^3 e x^2+148680 a^2 c^2 d^2 e^2 x^3+76800 a^2 c^2 d e^3 x^4+15624 a^2 c^2 e^4 x^5+43680 a c^3 d^4 x^3+138240 a c^3 d^3 e x^4+171360 a c^3 d^2 e^2 x^5+97280 a c^3 d e^3 x^6+21168 a c^3 e^4 x^7+13440 c^4 d^4 x^5+46080 c^4 d^3 e x^6+60480 c^4 d^2 e^2 x^7+35840 c^4 d e^3 x^8+8064 c^4 e^4 x^9\right )}{80640 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 692, normalized size = 2.25 \begin {gather*} \left [\frac {315 \, {\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (8064 \, c^{5} e^{4} x^{9} + 35840 \, c^{5} d e^{3} x^{8} + 46080 \, a^{3} c^{2} d^{3} e - 10240 \, a^{4} c d e^{3} + 3024 \, {\left (20 \, c^{5} d^{2} e^{2} + 7 \, a c^{4} e^{4}\right )} x^{7} + 5120 \, {\left (9 \, c^{5} d^{3} e + 19 \, a c^{4} d e^{3}\right )} x^{6} + 168 \, {\left (80 \, c^{5} d^{4} + 1020 \, a c^{4} d^{2} e^{2} + 93 \, a^{2} c^{3} e^{4}\right )} x^{5} + 15360 \, {\left (9 \, a c^{4} d^{3} e + 5 \, a^{2} c^{3} d e^{3}\right )} x^{4} + 210 \, {\left (208 \, a c^{4} d^{4} + 708 \, a^{2} c^{3} d^{2} e^{2} + 3 \, a^{3} c^{2} e^{4}\right )} x^{3} + 5120 \, {\left (27 \, a^{2} c^{3} d^{3} e + a^{3} c^{2} d e^{3}\right )} x^{2} + 315 \, {\left (176 \, a^{2} c^{3} d^{4} + 60 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{161280 \, c^{3}}, -\frac {315 \, {\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (8064 \, c^{5} e^{4} x^{9} + 35840 \, c^{5} d e^{3} x^{8} + 46080 \, a^{3} c^{2} d^{3} e - 10240 \, a^{4} c d e^{3} + 3024 \, {\left (20 \, c^{5} d^{2} e^{2} + 7 \, a c^{4} e^{4}\right )} x^{7} + 5120 \, {\left (9 \, c^{5} d^{3} e + 19 \, a c^{4} d e^{3}\right )} x^{6} + 168 \, {\left (80 \, c^{5} d^{4} + 1020 \, a c^{4} d^{2} e^{2} + 93 \, a^{2} c^{3} e^{4}\right )} x^{5} + 15360 \, {\left (9 \, a c^{4} d^{3} e + 5 \, a^{2} c^{3} d e^{3}\right )} x^{4} + 210 \, {\left (208 \, a c^{4} d^{4} + 708 \, a^{2} c^{3} d^{2} e^{2} + 3 \, a^{3} c^{2} e^{4}\right )} x^{3} + 5120 \, {\left (27 \, a^{2} c^{3} d^{3} e + a^{3} c^{2} d e^{3}\right )} x^{2} + 315 \, {\left (176 \, a^{2} c^{3} d^{4} + 60 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{80640 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 360, normalized size = 1.17 \begin {gather*} \frac {1}{80640} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, c^{2} x e^{4} + 40 \, c^{2} d e^{3}\right )} x + \frac {27 \, {\left (20 \, c^{10} d^{2} e^{2} + 7 \, a c^{9} e^{4}\right )}}{c^{8}}\right )} x + \frac {320 \, {\left (9 \, c^{10} d^{3} e + 19 \, a c^{9} d e^{3}\right )}}{c^{8}}\right )} x + \frac {21 \, {\left (80 \, c^{10} d^{4} + 1020 \, a c^{9} d^{2} e^{2} + 93 \, a^{2} c^{8} e^{4}\right )}}{c^{8}}\right )} x + \frac {1920 \, {\left (9 \, a c^{9} d^{3} e + 5 \, a^{2} c^{8} d e^{3}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (208 \, a c^{9} d^{4} + 708 \, a^{2} c^{8} d^{2} e^{2} + 3 \, a^{3} c^{7} e^{4}\right )}}{c^{8}}\right )} x + \frac {2560 \, {\left (27 \, a^{2} c^{8} d^{3} e + a^{3} c^{7} d e^{3}\right )}}{c^{8}}\right )} x + \frac {315 \, {\left (176 \, a^{2} c^{8} d^{4} + 60 \, a^{3} c^{7} d^{2} e^{2} - 3 \, a^{4} c^{6} e^{4}\right )}}{c^{8}}\right )} x + \frac {5120 \, {\left (9 \, a^{3} c^{7} d^{3} e - 2 \, a^{4} c^{6} d e^{3}\right )}}{c^{8}}\right )} - \frac {{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{256 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 386, normalized size = 1.26 \begin {gather*} \frac {3 a^{5} e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{256 c^{\frac {5}{2}}}-\frac {15 a^{4} d^{2} e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{64 c^{\frac {3}{2}}}+\frac {5 a^{3} d^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+a}\, a^{4} e^{4} x}{256 c^{2}}-\frac {15 \sqrt {c \,x^{2}+a}\, a^{3} d^{2} e^{2} x}{64 c}+\frac {5 \sqrt {c \,x^{2}+a}\, a^{2} d^{4} x}{16}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{3} e^{4} x}{128 c^{2}}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} d^{2} e^{2} x}{32 c}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,d^{4} x}{24}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} e^{4} x^{3}}{10 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} a^{2} e^{4} x}{160 c^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} a \,d^{2} e^{2} x}{8 c}+\frac {4 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d \,e^{3} x^{2}}{9 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} d^{4} x}{6}-\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a \,e^{4} x}{80 c^{2}}+\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d^{2} e^{2} x}{4 c}-\frac {8 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a d \,e^{3}}{63 c^{2}}+\frac {4 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d^{3} e}{7 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 364, normalized size = 1.19 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{4} x^{3}}{10 \, c} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e^{3} x^{2}}{9 \, c} + \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{4} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{4} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{4} x + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{2} e^{2} x}{4 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a d^{2} e^{2} x}{8 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2} e^{2} x}{32 \, c} - \frac {15 \, \sqrt {c x^{2} + a} a^{3} d^{2} e^{2} x}{64 \, c} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a e^{4} x}{80 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2} e^{4} x}{160 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{4} x}{128 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{4} e^{4} x}{256 \, c^{2}} + \frac {5 \, a^{3} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {15 \, a^{4} d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{64 \, c^{\frac {3}{2}}} + \frac {3 \, a^{5} e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{256 \, c^{\frac {5}{2}}} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{3} e}{7 \, c} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a d e^{3}}{63 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 63.41, size = 1062, normalized size = 3.46
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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