3.1446 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=346 \[ \frac{2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^{3/2}}+\frac{6 c^2 \sqrt{d+e x} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{2 c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^{7/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^{9/2}}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{3/2} (7 B d-A e)}{3 e^8}+\frac{2 B c^3 (d+e x)^{5/2}}{5 e^8} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(9*e^8*(d + e*x)^(9/2)) - (2*(c*d^2 + a*e^2)^2
*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^(7/2)) + (6*c*(c*d^2 + a*e^
2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^8*(d + e*x)^(5/2)) +
(2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4
)))/(3*e^8*(d + e*x)^(3/2)) + (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 -
3*a*A*e^3))/(e^8*Sqrt[d + e*x]) + (6*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Sqrt
[d + e*x])/e^8 - (2*c^3*(7*B*d - A*e)*(d + e*x)^(3/2))/(3*e^8) + (2*B*c^3*(d + e
*x)^(5/2))/(5*e^8)

_______________________________________________________________________________________

Rubi [A]  time = 0.459511, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{2 c \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{3 e^8 (d+e x)^{3/2}}+\frac{6 c^2 \sqrt{d+e x} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{2 c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^{7/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^{9/2}}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{3/2} (7 B d-A e)}{3 e^8}+\frac{2 B c^3 (d+e x)^{5/2}}{5 e^8} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(9*e^8*(d + e*x)^(9/2)) - (2*(c*d^2 + a*e^2)^2
*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^(7/2)) + (6*c*(c*d^2 + a*e^
2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^8*(d + e*x)^(5/2)) -
(2*c*(35*B*c^2*d^4 - 20*A*c^2*d^3*e + 30*a*B*c*d^2*e^2 - 12*a*A*c*d*e^3 + 3*a^2*
B*e^4))/(3*e^8*(d + e*x)^(3/2)) + (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e
^2 - 3*a*A*e^3))/(e^8*Sqrt[d + e*x]) + (6*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*
Sqrt[d + e*x])/e^8 - (2*c^3*(7*B*d - A*e)*(d + e*x)^(3/2))/(3*e^8) + (2*B*c^3*(d
+ e*x)^(5/2))/(5*e^8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 106.315, size = 364, normalized size = 1.05 \[ \frac{2 B c^{3} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{8}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{3}{2}} \left (A e - 7 B d\right )}{3 e^{8}} + \frac{6 c^{2} \sqrt{d + e x} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{8}} - \frac{2 c^{2} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{e^{8} \sqrt{d + e x}} - \frac{2 c \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{3 e^{8} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{5 e^{8} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{7 e^{8} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{9 e^{8} \left (d + e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(11/2),x)

[Out]

2*B*c**3*(d + e*x)**(5/2)/(5*e**8) + 2*c**3*(d + e*x)**(3/2)*(A*e - 7*B*d)/(3*e*
*8) + 6*c**2*sqrt(d + e*x)*(-2*A*c*d*e + B*a*e**2 + 7*B*c*d**2)/e**8 - 2*c**2*(3
*A*a*e**3 + 15*A*c*d**2*e - 15*B*a*d*e**2 - 35*B*c*d**3)/(e**8*sqrt(d + e*x)) -
2*c*(-12*A*a*c*d*e**3 - 20*A*c**2*d**3*e + 3*B*a**2*e**4 + 30*B*a*c*d**2*e**2 +
35*B*c**2*d**4)/(3*e**8*(d + e*x)**(3/2)) - 6*c*(a*e**2 + c*d**2)*(A*a*e**3 + 5*
A*c*d**2*e - 3*B*a*d*e**2 - 7*B*c*d**3)/(5*e**8*(d + e*x)**(5/2)) - 2*(a*e**2 +
c*d**2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c*d**2)/(7*e**8*(d + e*x)**(7/2)) - 2*(A
*e - B*d)*(a*e**2 + c*d**2)**3/(9*e**8*(d + e*x)**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 2.3122, size = 300, normalized size = 0.87 \[ \frac{2 \sqrt{d+e x} \left (-\frac{105 c \left (B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )-4 A c d e \left (3 a e^2+5 c d^2\right )\right )}{(d+e x)^2}+21 c^2 \left (45 a B e^2-85 A c d e+283 B c d^2\right )+\frac{315 c^2 \left (5 B \left (3 a d e^2+7 c d^3\right )-3 A e \left (a e^2+5 c d^2\right )\right )}{d+e x}-\frac{45 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{(d+e x)^4}+\frac{35 \left (a e^2+c d^2\right )^3 (B d-A e)}{(d+e x)^5}+\frac{189 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{(d+e x)^3}+21 c^3 e x (5 A e-29 B d)+63 B c^3 e^2 x^2\right )}{315 e^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[d + e*x]*(21*c^2*(283*B*c*d^2 - 85*A*c*d*e + 45*a*B*e^2) + 21*c^3*e*(-29
*B*d + 5*A*e)*x + 63*B*c^3*e^2*x^2 + (35*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(d + e*x
)^5 - (45*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(d + e*x)^4 + (18
9*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(d + e*x)
^3 - (105*c*(-4*A*c*d*e*(5*c*d^2 + 3*a*e^2) + B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3
*a^2*e^4)))/(d + e*x)^2 + (315*c^2*(-3*A*e*(5*c*d^2 + a*e^2) + 5*B*(7*c*d^3 + 3*
a*d*e^2)))/(d + e*x)))/(315*e^8)

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 489, normalized size = 1.4 \[ -{\frac{-126\,B{c}^{3}{x}^{7}{e}^{7}-210\,A{c}^{3}{e}^{7}{x}^{6}+588\,B{c}^{3}d{e}^{6}{x}^{6}+2520\,A{c}^{3}d{e}^{6}{x}^{5}-1890\,Ba{c}^{2}{e}^{7}{x}^{5}-7056\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}+1890\,Aa{c}^{2}{e}^{7}{x}^{4}+25200\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}-18900\,Ba{c}^{2}d{e}^{6}{x}^{4}-70560\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+5040\,Aa{c}^{2}d{e}^{6}{x}^{3}+67200\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}+630\,B{a}^{2}c{e}^{7}{x}^{3}-50400\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-188160\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}+378\,A{a}^{2}c{e}^{7}{x}^{2}+6048\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}+80640\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}+756\,B{a}^{2}cd{e}^{6}{x}^{2}-60480\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}-225792\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+216\,A{a}^{2}cd{e}^{6}x+3456\,Aa{c}^{2}{d}^{3}{e}^{4}x+46080\,A{c}^{3}{d}^{5}{e}^{2}x+90\,B{a}^{3}{e}^{7}x+432\,B{a}^{2}c{d}^{2}{e}^{5}x-34560\,Ba{c}^{2}{d}^{4}{e}^{3}x-129024\,B{c}^{3}{d}^{6}ex+70\,A{a}^{3}{e}^{7}+48\,A{a}^{2}c{d}^{2}{e}^{5}+768\,Aa{c}^{2}{d}^{4}{e}^{3}+10240\,A{c}^{3}{d}^{6}e+20\,B{a}^{3}d{e}^{6}+96\,B{a}^{2}c{d}^{3}{e}^{4}-7680\,Ba{c}^{2}{d}^{5}{e}^{2}-28672\,B{c}^{3}{d}^{7}}{315\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x)

[Out]

-2/315/(e*x+d)^(9/2)*(-63*B*c^3*e^7*x^7-105*A*c^3*e^7*x^6+294*B*c^3*d*e^6*x^6+12
60*A*c^3*d*e^6*x^5-945*B*a*c^2*e^7*x^5-3528*B*c^3*d^2*e^5*x^5+945*A*a*c^2*e^7*x^
4+12600*A*c^3*d^2*e^5*x^4-9450*B*a*c^2*d*e^6*x^4-35280*B*c^3*d^3*e^4*x^4+2520*A*
a*c^2*d*e^6*x^3+33600*A*c^3*d^3*e^4*x^3+315*B*a^2*c*e^7*x^3-25200*B*a*c^2*d^2*e^
5*x^3-94080*B*c^3*d^4*e^3*x^3+189*A*a^2*c*e^7*x^2+3024*A*a*c^2*d^2*e^5*x^2+40320
*A*c^3*d^4*e^3*x^2+378*B*a^2*c*d*e^6*x^2-30240*B*a*c^2*d^3*e^4*x^2-112896*B*c^3*
d^5*e^2*x^2+108*A*a^2*c*d*e^6*x+1728*A*a*c^2*d^3*e^4*x+23040*A*c^3*d^5*e^2*x+45*
B*a^3*e^7*x+216*B*a^2*c*d^2*e^5*x-17280*B*a*c^2*d^4*e^3*x-64512*B*c^3*d^6*e*x+35
*A*a^3*e^7+24*A*a^2*c*d^2*e^5+384*A*a*c^2*d^4*e^3+5120*A*c^3*d^6*e+10*B*a^3*d*e^
6+48*B*a^2*c*d^3*e^4-3840*B*a*c^2*d^5*e^2-14336*B*c^3*d^7)/e^8

_______________________________________________________________________________________

Maxima [A]  time = 0.690897, size = 622, normalized size = 1.8 \[ \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c^{3} - 5 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 45 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \sqrt{e x + d}\right )}}{e^{7}} + \frac{35 \, B c^{3} d^{7} - 35 \, A c^{3} d^{6} e + 105 \, B a c^{2} d^{5} e^{2} - 105 \, A a c^{2} d^{4} e^{3} + 105 \, B a^{2} c d^{3} e^{4} - 105 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} + 315 \,{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )}{\left (e x + d\right )}^{4} - 105 \,{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )}{\left (e x + d\right )}^{3} + 189 \,{\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )}{\left (e x + d\right )}^{2} - 45 \,{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{9}{2}} e^{7}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

2/315*(21*(3*(e*x + d)^(5/2)*B*c^3 - 5*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(3/2) + 4
5*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*sqrt(e*x + d))/e^7 + (35*B*c^3*d^7 -
35*A*c^3*d^6*e + 105*B*a*c^2*d^5*e^2 - 105*A*a*c^2*d^4*e^3 + 105*B*a^2*c*d^3*e^
4 - 105*A*a^2*c*d^2*e^5 + 35*B*a^3*d*e^6 - 35*A*a^3*e^7 + 315*(35*B*c^3*d^3 - 15
*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*(e*x + d)^4 - 105*(35*B*c^3*d^4
- 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*(e*x
+ d)^3 + 189*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e
^3 + 3*B*a^2*c*d*e^4 - A*a^2*c*e^5)*(e*x + d)^2 - 45*(7*B*c^3*d^6 - 6*A*c^3*d^5*
e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^
5 + B*a^3*e^6)*(e*x + d))/((e*x + d)^(9/2)*e^7))/e

_______________________________________________________________________________________

Fricas [A]  time = 0.268044, size = 672, normalized size = 1.94 \[ \frac{2 \,{\left (63 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 5120 \, A c^{3} d^{6} e + 3840 \, B a c^{2} d^{5} e^{2} - 384 \, A a c^{2} d^{4} e^{3} - 48 \, B a^{2} c d^{3} e^{4} - 24 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} - 21 \,{\left (14 \, B c^{3} d e^{6} - 5 \, A c^{3} e^{7}\right )} x^{6} + 63 \,{\left (56 \, B c^{3} d^{2} e^{5} - 20 \, A c^{3} d e^{6} + 15 \, B a c^{2} e^{7}\right )} x^{5} + 315 \,{\left (112 \, B c^{3} d^{3} e^{4} - 40 \, A c^{3} d^{2} e^{5} + 30 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 105 \,{\left (896 \, B c^{3} d^{4} e^{3} - 320 \, A c^{3} d^{3} e^{4} + 240 \, B a c^{2} d^{2} e^{5} - 24 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 63 \,{\left (1792 \, B c^{3} d^{5} e^{2} - 640 \, A c^{3} d^{4} e^{3} + 480 \, B a c^{2} d^{3} e^{4} - 48 \, A a c^{2} d^{2} e^{5} - 6 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 9 \,{\left (7168 \, B c^{3} d^{6} e - 2560 \, A c^{3} d^{5} e^{2} + 1920 \, B a c^{2} d^{4} e^{3} - 192 \, A a c^{2} d^{3} e^{4} - 24 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 5 \, B a^{3} e^{7}\right )} x\right )}}{315 \,{\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*c^3*e^7*x^7 + 14336*B*c^3*d^7 - 5120*A*c^3*d^6*e + 3840*B*a*c^2*d^5*
e^2 - 384*A*a*c^2*d^4*e^3 - 48*B*a^2*c*d^3*e^4 - 24*A*a^2*c*d^2*e^5 - 10*B*a^3*d
*e^6 - 35*A*a^3*e^7 - 21*(14*B*c^3*d*e^6 - 5*A*c^3*e^7)*x^6 + 63*(56*B*c^3*d^2*e
^5 - 20*A*c^3*d*e^6 + 15*B*a*c^2*e^7)*x^5 + 315*(112*B*c^3*d^3*e^4 - 40*A*c^3*d^
2*e^5 + 30*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 105*(896*B*c^3*d^4*e^3 - 320*A*c
^3*d^3*e^4 + 240*B*a*c^2*d^2*e^5 - 24*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 63*(1
792*B*c^3*d^5*e^2 - 640*A*c^3*d^4*e^3 + 480*B*a*c^2*d^3*e^4 - 48*A*a*c^2*d^2*e^5
- 6*B*a^2*c*d*e^6 - 3*A*a^2*c*e^7)*x^2 + 9*(7168*B*c^3*d^6*e - 2560*A*c^3*d^5*e
^2 + 1920*B*a*c^2*d^4*e^3 - 192*A*a*c^2*d^3*e^4 - 24*B*a^2*c*d^2*e^5 - 12*A*a^2*
c*d*e^6 - 5*B*a^3*e^7)*x)/((e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9
*x + d^4*e^8)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [A]  time = 54.0122, size = 3952, normalized size = 11.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(11/2),x)

[Out]

Piecewise((-70*A*a**3*e**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(
d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x)
+ 315*e**12*x**4*sqrt(d + e*x)) - 48*A*a**2*c*d**2*e**5/(315*d**4*e**8*sqrt(d +
e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 126
0*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 216*A*a**2*c*d*e*
*6*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e
**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(
d + e*x)) - 378*A*a**2*c*e**7*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9
*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d
+ e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 768*A*a*c**2*d**4*e**3/(315*d**4*e**8*
sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e
*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 3456*A*a
*c**2*d**3*e**4*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x)
+ 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**
12*x**4*sqrt(d + e*x)) - 6048*A*a*c**2*d**2*e**5*x**2/(315*d**4*e**8*sqrt(d + e*
x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*
d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 5040*A*a*c**2*d*e**
6*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2
*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqr
t(d + e*x)) - 1890*A*a*c**2*e**7*x**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e
**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqr
t(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 10240*A*c**3*d**6*e/(315*d**4*e**8*
sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e
*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 46080*A*
c**3*d**5*e**2*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) +
1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**1
2*x**4*sqrt(d + e*x)) - 80640*A*c**3*d**4*e**3*x**2/(315*d**4*e**8*sqrt(d + e*x)
+ 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*
e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 67200*A*c**3*d**3*e**
4*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2
*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqr
t(d + e*x)) - 25200*A*c**3*d**2*e**5*x**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d*
*3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3
*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 2520*A*c**3*d*e**6*x**5/(315*d*
*4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sq
rt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) +
210*A*c**3*e**7*x**6/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*
x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*
e**12*x**4*sqrt(d + e*x)) - 20*B*a**3*d*e**6/(315*d**4*e**8*sqrt(d + e*x) + 1260
*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x
**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 90*B*a**3*e**7*x/(315*d**4*e
**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d
+ e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 96*B
*a**2*c*d**3*e**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x)
+ 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**
12*x**4*sqrt(d + e*x)) - 432*B*a**2*c*d**2*e**5*x/(315*d**4*e**8*sqrt(d + e*x) +
1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e*
*11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 756*B*a**2*c*d*e**6*x**
2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**1
0*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d +
e*x)) - 630*B*a**2*c*e**7*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*
sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d +
e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 7680*B*a*c**2*d**5*e**2/(315*d**4*e**8*sq
rt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x
) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 34560*B*a*
c**2*d**4*e**3*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) +
1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**1
2*x**4*sqrt(d + e*x)) + 60480*B*a*c**2*d**3*e**4*x**2/(315*d**4*e**8*sqrt(d + e*
x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*
d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 50400*B*a*c**2*d**2
*e**5*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*
d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4
*sqrt(d + e*x)) + 18900*B*a*c**2*d*e**6*x**4/(315*d**4*e**8*sqrt(d + e*x) + 1260
*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x
**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 1890*B*a*c**2*e**7*x**5/(315
*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2
*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x))
+ 28672*B*c**3*d**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*
x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*
e**12*x**4*sqrt(d + e*x)) + 129024*B*c**3*d**6*e*x/(315*d**4*e**8*sqrt(d + e*x)
+ 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e
**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 225792*B*c**3*d**5*e**
2*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2
*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqr
t(d + e*x)) + 188160*B*c**3*d**4*e**3*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d
**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**
3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 70560*B*c**3*d**3*e**4*x**4/(3
15*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x*
*2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x
)) + 7056*B*c**3*d**2*e**5*x**5/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*
sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d +
e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 588*B*c**3*d*e**6*x**6/(315*d**4*e**8*sqr
t(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x)
+ 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 126*B*c**3*
e**7*x**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d
**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*
sqrt(d + e*x)), Ne(e, 0)), ((A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c*
*3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 + B*c**3*x**8/8)
/d**(11/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.32612, size = 803, normalized size = 2.32 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{3} e^{32} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{3} d e^{32} + 315 \, \sqrt{x e + d} B c^{3} d^{2} e^{32} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} e^{33} - 90 \, \sqrt{x e + d} A c^{3} d e^{33} + 45 \, \sqrt{x e + d} B a c^{2} e^{34}\right )} e^{\left (-40\right )} + \frac{2 \,{\left (11025 \,{\left (x e + d\right )}^{4} B c^{3} d^{3} - 3675 \,{\left (x e + d\right )}^{3} B c^{3} d^{4} + 1323 \,{\left (x e + d\right )}^{2} B c^{3} d^{5} - 315 \,{\left (x e + d\right )} B c^{3} d^{6} + 35 \, B c^{3} d^{7} - 4725 \,{\left (x e + d\right )}^{4} A c^{3} d^{2} e + 2100 \,{\left (x e + d\right )}^{3} A c^{3} d^{3} e - 945 \,{\left (x e + d\right )}^{2} A c^{3} d^{4} e + 270 \,{\left (x e + d\right )} A c^{3} d^{5} e - 35 \, A c^{3} d^{6} e + 4725 \,{\left (x e + d\right )}^{4} B a c^{2} d e^{2} - 3150 \,{\left (x e + d\right )}^{3} B a c^{2} d^{2} e^{2} + 1890 \,{\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} - 675 \,{\left (x e + d\right )} B a c^{2} d^{4} e^{2} + 105 \, B a c^{2} d^{5} e^{2} - 945 \,{\left (x e + d\right )}^{4} A a c^{2} e^{3} + 1260 \,{\left (x e + d\right )}^{3} A a c^{2} d e^{3} - 1134 \,{\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} + 540 \,{\left (x e + d\right )} A a c^{2} d^{3} e^{3} - 105 \, A a c^{2} d^{4} e^{3} - 315 \,{\left (x e + d\right )}^{3} B a^{2} c e^{4} + 567 \,{\left (x e + d\right )}^{2} B a^{2} c d e^{4} - 405 \,{\left (x e + d\right )} B a^{2} c d^{2} e^{4} + 105 \, B a^{2} c d^{3} e^{4} - 189 \,{\left (x e + d\right )}^{2} A a^{2} c e^{5} + 270 \,{\left (x e + d\right )} A a^{2} c d e^{5} - 105 \, A a^{2} c d^{2} e^{5} - 45 \,{\left (x e + d\right )} B a^{3} e^{6} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{315 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c^3*e^32 - 35*(x*e + d)^(3/2)*B*c^3*d*e^32 + 315*sqrt(
x*e + d)*B*c^3*d^2*e^32 + 5*(x*e + d)^(3/2)*A*c^3*e^33 - 90*sqrt(x*e + d)*A*c^3*
d*e^33 + 45*sqrt(x*e + d)*B*a*c^2*e^34)*e^(-40) + 2/315*(11025*(x*e + d)^4*B*c^3
*d^3 - 3675*(x*e + d)^3*B*c^3*d^4 + 1323*(x*e + d)^2*B*c^3*d^5 - 315*(x*e + d)*B
*c^3*d^6 + 35*B*c^3*d^7 - 4725*(x*e + d)^4*A*c^3*d^2*e + 2100*(x*e + d)^3*A*c^3*
d^3*e - 945*(x*e + d)^2*A*c^3*d^4*e + 270*(x*e + d)*A*c^3*d^5*e - 35*A*c^3*d^6*e
+ 4725*(x*e + d)^4*B*a*c^2*d*e^2 - 3150*(x*e + d)^3*B*a*c^2*d^2*e^2 + 1890*(x*e
+ d)^2*B*a*c^2*d^3*e^2 - 675*(x*e + d)*B*a*c^2*d^4*e^2 + 105*B*a*c^2*d^5*e^2 -
945*(x*e + d)^4*A*a*c^2*e^3 + 1260*(x*e + d)^3*A*a*c^2*d*e^3 - 1134*(x*e + d)^2*
A*a*c^2*d^2*e^3 + 540*(x*e + d)*A*a*c^2*d^3*e^3 - 105*A*a*c^2*d^4*e^3 - 315*(x*e
+ d)^3*B*a^2*c*e^4 + 567*(x*e + d)^2*B*a^2*c*d*e^4 - 405*(x*e + d)*B*a^2*c*d^2*
e^4 + 105*B*a^2*c*d^3*e^4 - 189*(x*e + d)^2*A*a^2*c*e^5 + 270*(x*e + d)*A*a^2*c*
d*e^5 - 105*A*a^2*c*d^2*e^5 - 45*(x*e + d)*B*a^3*e^6 + 35*B*a^3*d*e^6 - 35*A*a^3
*e^7)*e^(-8)/(x*e + d)^(9/2)