Optimal. Leaf size=204 \[ \frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711}
\begin {gather*} \frac {6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac {8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac {2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{e^6}-\frac {6 c^3 d (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 170, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (51051 a^3 e^6+2431 a^2 c e^4 \left (8 d^2-20 d e x+35 e^2 x^2\right )+51 a c^2 e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 269, normalized size = 1.32
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 c^{3} x^{6} e^{6}-12012 c^{3} d \,e^{5} x^{5}+58905 a \,c^{2} e^{6} x^{4}+9240 c^{3} d^{2} e^{4} x^{4}-42840 a \,c^{2} d \,e^{5} x^{3}-6720 c^{3} d^{3} e^{3} x^{3}+85085 a^{2} c \,e^{6} x^{2}+28560 a \,c^{2} d^{2} e^{4} x^{2}+4480 c^{3} d^{4} e^{2} x^{2}-48620 a^{2} c d \,e^{5} x -16320 a \,c^{2} d^{3} e^{3} x -2560 c^{3} d^{5} e x +51051 e^{6} a^{3}+19448 e^{4} d^{2} a^{2} c +6528 d^{4} e^{2} c^{2} a +1024 d^{6} c^{3}\right )}{255255 e^{7}}\) | \(205\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
trager | \(\frac {2 \left (15015 c^{3} e^{8} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 d \,e^{7} a \,c^{2} x^{5}-252 e^{5} d^{3} c^{3} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
risch | \(\frac {2 \left (15015 c^{3} e^{8} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 d \,e^{7} a \,c^{2} x^{5}-252 e^{5} d^{3} c^{3} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 206, normalized size = 1.01 \begin {gather*} \frac {2}{255255} \, {\left (15015 \, {\left (x e + d\right )}^{\frac {17}{2}} c^{3} - 102102 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{3} d + 58905 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 92820 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 218790 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.56, size = 285, normalized size = 1.40 \begin {gather*} -\frac {2}{255255} \, {\left (512 \, c^{3} d^{7} x e - 1024 \, c^{3} d^{8} - 77 \, {\left (195 \, c^{3} x^{8} + 765 \, a c^{2} x^{6} + 1105 \, a^{2} c x^{4} + 663 \, a^{3} x^{2}\right )} e^{8} - 2 \, {\left (9009 \, c^{3} d x^{7} + 37485 \, a c^{2} d x^{5} + 60775 \, a^{2} c d x^{3} + 51051 \, a^{3} d x\right )} e^{7} - 3 \, {\left (77 \, c^{3} d^{2} x^{6} + 595 \, a c^{2} d^{2} x^{4} + 2431 \, a^{2} c d^{2} x^{2} + 17017 \, a^{3} d^{2}\right )} e^{6} + 4 \, {\left (63 \, c^{3} d^{3} x^{5} + 510 \, a c^{2} d^{3} x^{3} + 2431 \, a^{2} c d^{3} x\right )} e^{5} - 8 \, {\left (35 \, c^{3} d^{4} x^{4} + 306 \, a c^{2} d^{4} x^{2} + 2431 \, a^{2} c d^{4}\right )} e^{4} + 64 \, {\left (5 \, c^{3} d^{5} x^{3} + 51 \, a c^{2} d^{5} x\right )} e^{3} - 384 \, {\left (c^{3} d^{6} x^{2} + 17 \, a c^{2} d^{6}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.53, size = 564, normalized size = 2.76 \begin {gather*} a^{3} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{3} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {6 a^{2} c d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {6 a^{2} c \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {6 a c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {6 a c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} + \frac {2 c^{3} d \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{7}} + \frac {2 c^{3} \left (- \frac {d^{7} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {7 d^{6} \left (d + e x\right )^{\frac {5}{2}}}{5} - 3 d^{5} \left (d + e x\right )^{\frac {7}{2}} + \frac {35 d^{4} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {35 d^{3} \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {21 d^{2} \left (d + e x\right )^{\frac {13}{2}}}{13} - \frac {7 d \left (d + e x\right )^{\frac {15}{2}}}{15} + \frac {\left (d + e x\right )^{\frac {17}{2}}}{17}\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 834 vs.
\(2 (170) = 340\).
time = 4.32, size = 834, normalized size = 4.09 \begin {gather*} \frac {2}{765765} \, {\left (153153 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} c d^{2} e^{\left (-2\right )} + 7293 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c^{2} d^{2} e^{\left (-4\right )} + 255 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} d^{2} e^{\left (-6\right )} + 131274 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} c d e^{\left (-2\right )} + 6630 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a c^{2} d e^{\left (-4\right )} + 238 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} d e^{\left (-6\right )} + 765765 \, \sqrt {x e + d} a^{3} d^{2} + 510510 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d + 7293 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} c e^{\left (-2\right )} + 765 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a c^{2} e^{\left (-4\right )} + 7 \, {\left (6435 \, {\left (x e + d\right )}^{\frac {17}{2}} - 58344 \, {\left (x e + d\right )}^{\frac {15}{2}} d + 235620 \, {\left (x e + d\right )}^{\frac {13}{2}} d^{2} - 556920 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{3} + 850850 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{4} - 875160 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{5} + 612612 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{6} - 291720 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{7} + 109395 \, \sqrt {x e + d} d^{8}\right )} c^{3} e^{\left (-6\right )} + 51051 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 187, normalized size = 0.92 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{15/2}}{5\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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