3.4.53 \(\int (d+e x)^{7/2} (b x+c x^2)^3 \, dx\) [353]
Optimal. Leaf size=248 \[ \frac {2 d^3 (c d-b e)^3 (d+e x)^{9/2}}{9 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{15 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{17/2}}{17 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{19/2}}{19 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7} \]
[Out]
2/9*d^3*(-b*e+c*d)^3*(e*x+d)^(9/2)/e^7-6/11*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^7+6/13*d*(-b*e+c*d)
*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(13/2)/e^7-2/15*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(1
5/2)/e^7+6/17*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(17/2)/e^7-6/19*c^2*(-b*e+2*c*d)*(e*x+d)^(19/2)/e^7+2/21
*c^3*(e*x+d)^(21/2)/e^7
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Rubi [A]
time = 0.10, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {712}
\begin {gather*} \frac {6 c (d+e x)^{17/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{17 e^7}-\frac {2 (d+e x)^{15/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{15 e^7}+\frac {6 d (d+e x)^{13/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}+\frac {2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}-\frac {6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]
[Out]
(2*d^3*(c*d - b*e)^3*(d + e*x)^(9/2))/(9*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7)
+ (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2
- 10*b*c*d*e + b^2*e^2)*(d + e*x)^(15/2))/(15*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(17/2))
/(17*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(19/2))/(19*e^7) + (2*c^3*(d + e*x)^(21/2))/(21*e^7)
Rule 712
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin {align*} \int (d+e x)^{7/2} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 (d+e x)^{7/2}}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{9/2}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{13/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{17/2}}{e^6}+\frac {c^3 (d+e x)^{19/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 (d+e x)^{9/2}}{9 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{15 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{17/2}}{17 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{19/2}}{19 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 232, normalized size = 0.94 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (2261 b^3 e^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+399 b^2 c e^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+105 b c^2 e \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+5 c^3 \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )\right )}{14549535 e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(2261*b^3*e^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 399*b^2*c*e^2*(128*d^4
- 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) + 105*b*c^2*e*(-256*d^5 + 1152*d^4*e*x - 31
68*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 - 12870*d*e^4*x^4 + 21879*e^5*x^5) + 5*c^3*(1024*d^6 - 4608*d^5*e*x + 12672*
d^4*e^2*x^2 - 27456*d^3*e^3*x^3 + 51480*d^2*e^4*x^4 - 87516*d*e^5*x^5 + 138567*e^6*x^6)))/(14549535*e^7)
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Maple [A]
time = 0.43, size = 269, normalized size = 1.08 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(7/2)*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
[Out]
2/e^7*(1/21*c^3*(e*x+d)^(21/2)+1/19*(-3*c^3*d+3*(b*e-c*d)*c^2)*(e*x+d)^(19/2)+1/17*(3*c^3*d^2-9*d*(b*e-c*d)*c^
2+3*(b*e-c*d)^2*c)*(e*x+d)^(17/2)+1/15*(-c^3*d^3+9*d^2*(b*e-c*d)*c^2-9*d*(b*e-c*d)^2*c+(b*e-c*d)^3)*(e*x+d)^(1
5/2)+1/13*(-3*d^3*(b*e-c*d)*c^2+9*d^2*(b*e-c*d)^2*c-3*d*(b*e-c*d)^3)*(e*x+d)^(13/2)+1/11*(-3*d^3*(b*e-c*d)^2*c
+3*d^2*(b*e-c*d)^3)*(e*x+d)^(11/2)-1/9*d^3*(b*e-c*d)^3*(e*x+d)^(9/2))
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Maxima [A]
time = 0.28, size = 274, normalized size = 1.10 \begin {gather*} \frac {2}{14549535} \, {\left (692835 \, {\left (x e + d\right )}^{\frac {21}{2}} c^{3} - 2297295 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {19}{2}} + 2567565 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (x e + d\right )}^{\frac {17}{2}} - 969969 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 3357585 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 3968055 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 1616615 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="maxima")
[Out]
2/14549535*(692835*(x*e + d)^(21/2)*c^3 - 2297295*(2*c^3*d - b*c^2*e)*(x*e + d)^(19/2) + 2567565*(5*c^3*d^2 -
5*b*c^2*d*e + b^2*c*e^2)*(x*e + d)^(17/2) - 969969*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(x
*e + d)^(15/2) + 3357585*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(x*e + d)^(13/2) - 3968055
*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(x*e + d)^(11/2) + 1616615*(c^3*d^6 - 3*b*c^2*d^5
*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(x*e + d)^(9/2))*e^(-7)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs.
\(2 (226) = 452\).
time = 1.42, size = 460, normalized size = 1.85 \begin {gather*} \frac {2}{14549535} \, {\left (5120 \, c^{3} d^{10} + 429 \, {\left (1615 \, c^{3} x^{10} + 5355 \, b c^{2} x^{9} + 5985 \, b^{2} c x^{8} + 2261 \, b^{3} x^{7}\right )} e^{10} + 66 \, {\left (35360 \, c^{3} d x^{9} + 118755 \, b c^{2} d x^{8} + 134862 \, b^{2} c d x^{7} + 52003 \, b^{3} d x^{6}\right )} e^{9} + 18 \, {\left (148005 \, c^{3} d^{2} x^{8} + 505505 \, b c^{2} d^{2} x^{7} + 586663 \, b^{2} c d^{2} x^{6} + 232883 \, b^{3} d^{2} x^{5}\right )} e^{8} + 4 \, {\left (259545 \, c^{3} d^{3} x^{7} + 907830 \, b c^{2} d^{3} x^{6} + 1088073 \, b^{2} c d^{3} x^{5} + 452200 \, b^{3} d^{3} x^{4}\right )} e^{7} + 35 \, {\left (33 \, c^{3} d^{4} x^{6} + 189 \, b c^{2} d^{4} x^{5} + 399 \, b^{2} c d^{4} x^{4} + 323 \, b^{3} d^{4} x^{3}\right )} e^{6} - 42 \, {\left (30 \, c^{3} d^{5} x^{5} + 175 \, b c^{2} d^{5} x^{4} + 380 \, b^{2} c d^{5} x^{3} + 323 \, b^{3} d^{5} x^{2}\right )} e^{5} + 56 \, {\left (25 \, c^{3} d^{6} x^{4} + 150 \, b c^{2} d^{6} x^{3} + 342 \, b^{2} c d^{6} x^{2} + 323 \, b^{3} d^{6} x\right )} e^{4} - 16 \, {\left (100 \, c^{3} d^{7} x^{3} + 630 \, b c^{2} d^{7} x^{2} + 1596 \, b^{2} c d^{7} x + 2261 \, b^{3} d^{7}\right )} e^{3} + 384 \, {\left (5 \, c^{3} d^{8} x^{2} + 35 \, b c^{2} d^{8} x + 133 \, b^{2} c d^{8}\right )} e^{2} - 1280 \, {\left (2 \, c^{3} d^{9} x + 21 \, b c^{2} d^{9}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
2/14549535*(5120*c^3*d^10 + 429*(1615*c^3*x^10 + 5355*b*c^2*x^9 + 5985*b^2*c*x^8 + 2261*b^3*x^7)*e^10 + 66*(35
360*c^3*d*x^9 + 118755*b*c^2*d*x^8 + 134862*b^2*c*d*x^7 + 52003*b^3*d*x^6)*e^9 + 18*(148005*c^3*d^2*x^8 + 5055
05*b*c^2*d^2*x^7 + 586663*b^2*c*d^2*x^6 + 232883*b^3*d^2*x^5)*e^8 + 4*(259545*c^3*d^3*x^7 + 907830*b*c^2*d^3*x
^6 + 1088073*b^2*c*d^3*x^5 + 452200*b^3*d^3*x^4)*e^7 + 35*(33*c^3*d^4*x^6 + 189*b*c^2*d^4*x^5 + 399*b^2*c*d^4*
x^4 + 323*b^3*d^4*x^3)*e^6 - 42*(30*c^3*d^5*x^5 + 175*b*c^2*d^5*x^4 + 380*b^2*c*d^5*x^3 + 323*b^3*d^5*x^2)*e^5
+ 56*(25*c^3*d^6*x^4 + 150*b*c^2*d^6*x^3 + 342*b^2*c*d^6*x^2 + 323*b^3*d^6*x)*e^4 - 16*(100*c^3*d^7*x^3 + 630
*b*c^2*d^7*x^2 + 1596*b^2*c*d^7*x + 2261*b^3*d^7)*e^3 + 384*(5*c^3*d^8*x^2 + 35*b*c^2*d^8*x + 133*b^2*c*d^8)*e
^2 - 1280*(2*c^3*d^9*x + 21*b*c^2*d^9)*e)*sqrt(x*e + d)*e^(-7)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1741 vs.
\(2 (245) = 490\).
time = 28.54, size = 1741, normalized size = 7.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(7/2)*(c*x**2+b*x)**3,x)
[Out]
2*b**3*d**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/
9)/e**4 + 6*b**3*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(
d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 6*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) -
10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
4 + 2*b**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*
x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**4 + 6*b**2*c*
d**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9
+ (d + e*x)**(11/2)/11)/e**5 + 18*b**2*c*d**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d
+ e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 18*b**2
*c*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9
/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 6*b**2*c*(-d**7
*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d*
*3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**5
+ 6*b*c**2*d**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d +
e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 18*b*c**2*d**2*(d**6*(d + e*x)**(3/2)
/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**
(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 18*b*c**2*d*(-d**7*(d + e*x)**(3/2)/3 + 7*
d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11
+ 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 6*b*c**2*(d**8*(d + e
*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d +
e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d
+ e*x)**(19/2)/19)/e**6 + 2*c**3*d**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x
)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)*
*(15/2)/15)/e**7 + 6*c**3*d**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2)
+ 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(
15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 6*c**3*d*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6
*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 +
28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7 + 2*c**3*(-d**9*(d + e*x
)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d
+ e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2*(d + e*x)**(17/2)/1
7 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**7
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2065 vs.
\(2 (226) = 452\).
time = 1.44, size = 2065, normalized size = 8.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
2/14549535*(415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*
b^3*d^4*e^(-3) + 138567*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^
(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^2*c*d^4*e^(-4) + 62985*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990
*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b*c^2*d^4*
e^(-5) + 4845*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2
)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^3*d^4*e^(-6) + 184756*
(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e
+ d)*d^4)*b^3*d^3*e^(-3) + 251940*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 13
86*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^2*c*d^3*e^(-4) + 58140*(231*(x*e
+ d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(
5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b*c^2*d^3*e^(-5) + 9044*(429*(x*e + d)^(15/2) -
3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 -
27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*d^3*e^(-6) + 125970*(63*(
x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^
(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^3*d^2*e^(-3) + 87210*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 50
05*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003
*sqrt(x*e + d)*d^6)*b^2*c*d^2*e^(-4) + 40698*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)
^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e +
d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b*c^2*d^2*e^(-5) + 798*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)
*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)
^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*c^3*d^2*e^(-6
) + 19380*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^
3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^3*d*e^(-3) + 27132*(429*(x
*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e
+ d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^2*c*d*e^(-
4) + 1596*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(
11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e
+ d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b*c^2*d*e^(-5) + 252*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(1
7/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(
x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 -
230945*sqrt(x*e + d)*d^9)*c^3*d*e^(-6) + 2261*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d
)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e
+ d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^3*e^(-3) + 399*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d +
235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2
)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^2*c*e^(-4) + 189
*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*
d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e
+ d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*b*c^2*e^(-5) + 15*(46189*(x*e + d)^(21
/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d)^(17/2)*d^2 - 7759752*(x*e + d)^(15/2)*d^3 + 15668730*(x*e
+ d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 + 22632610*(x*e + d)^(9/2)*d^6 - 16628040*(x*e + d)^(7/2)*d^7
+ 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^(3/2)*d^9 + 969969*sqrt(x*e + d)*d^10)*c^3*e^(-6))*e^(-1)
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Mupad [B]
time = 0.23, size = 239, normalized size = 0.96 \begin {gather*} \frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{15\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{21/2}}{21\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {{\left (d+e\,x\right )}^{17/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{13\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x + c*x^2)^3*(d + e*x)^(7/2),x)
[Out]
((d + e*x)^(15/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(15*e^7) + (2*c^3*(d + e*x)^(21/
2))/(21*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(19/2))/(19*e^7) + ((d + e*x)^(17/2)*(30*c^3*d^2 + 6*b^2*c*e^
2 - 30*b*c^2*d*e))/(17*e^7) + ((d + e*x)^(13/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e)
)/(13*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(9/2))/(9*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(11/
2))/(11*e^7)
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