3.1.10 \(\int \frac {(a+b x)^3 (A+B x+C x^2+D x^3)}{(c+d x)^{3/2}} \, dx\) [10]

Optimal. Leaf size=434 \[ \frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \sqrt {c+d x}}{d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \]

[Out]

-2/3*(-a*d+b*c)*(a^2*d^2*(C*d-3*D*c)-a*b*d*(-3*B*d^2+8*C*c*d-15*D*c^2)+b^2*(3*A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*
c^3))*(d*x+c)^(3/2)/d^7+2/5*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*D*c)-3*a*b^2*d*(-B*d^2+4*C*c*d-10*D*c^2)+b^3*(A*d^3-
4*B*c*d^2+10*C*c^2*d-20*D*c^3))*(d*x+c)^(5/2)/d^7+2/7*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*D*c)-b^2*(-B*d^2+5*C*c*d-1
5*D*c^2))*(d*x+c)^(7/2)/d^7+2/9*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(9/2)/d^7+2/11*b^3*D*(d*x+c)^(11/2)/d^7+2*
(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^7/(d*x+c)^(1/2)-2*(-a*d+b*c)^2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(3
*A*d^3-4*B*c*d^2+5*C*c^2*d-6*D*c^3))*(d*x+c)^(1/2)/d^7

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1634} \begin {gather*} -\frac {2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac {2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}+\frac {2 (c+d x)^{5/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{5 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac {2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]

[Out]

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*Sqrt[c + d*x]) - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B
*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*Sqrt[c + d*x])/d^7 - (2*(b*c - a*d)*(a^2*d^2*
(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*(c +
d*x)^(3/2))/(3*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^
3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(5/2))/(5*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5
*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d
*x)^(9/2))/(9*d^7) + (2*b^3*D*(c + d*x)^(11/2))/(11*d^7)

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx &=\int \left (\frac {(-b c+a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^6 (c+d x)^{3/2}}+\frac {(b c-a d)^2 \left (-a d \left (2 c C d-B d^2-3 c^2 D\right )+b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right )}{d^6 \sqrt {c+d x}}+\frac {(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )-b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) \sqrt {c+d x}}{d^6}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{3/2}}{d^6}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{5/2}}{d^6}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{7/2}}{d^6}+\frac {b^3 D (c+d x)^{9/2}}{d^6}\right ) \, dx\\ &=\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \sqrt {c+d x}}{d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.60, size = 500, normalized size = 1.15 \begin {gather*} \frac {2 \left (231 a^3 d^3 \left (48 c^3 D-8 c^2 d (5 C-3 D x)+2 c d^2 (15 B-x (10 C+3 D x))+d^3 \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )\right )+99 a^2 b d^2 \left (-384 c^4 D+48 c^3 d (7 C-4 D x)-8 c^2 d^2 (35 B-3 x (7 C+2 D x))+2 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))\right )+33 a b^2 d \left (1280 c^5 D-128 c^4 d (9 C-5 D x)+16 c^3 d^2 (63 B-2 x (18 C+5 D x))+8 c^2 d^3 (-105 A+x (63 B+2 x (9 C+5 D x)))+d^5 x^2 (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^4 x (210 A+x (63 B+x (36 C+25 D x)))\right )+b^3 \left (-15360 c^6 D+1280 c^5 d (11 C-6 D x)-128 c^4 d^2 (99 B-5 x (11 C+3 D x))+16 c^3 d^3 (693 A-2 x (198 B+5 x (11 C+6 D x)))+d^6 x^3 (693 A+5 x (99 B+7 x (11 C+9 D x)))+8 c^2 d^4 x (693 A+x (198 B+5 x (22 C+15 D x)))-2 c d^5 x^2 (693 A+x (396 B+5 x (55 C+42 D x)))\right )\right )}{3465 d^7 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]

[Out]

(2*(231*a^3*d^3*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B +
5*C*x + 3*D*x^2))) + 99*a^2*b*d^2*(-384*c^4*D + 48*c^3*d*(7*C - 4*D*x) - 8*c^2*d^2*(35*B - 3*x*(7*C + 2*D*x))
+ 2*c*d^3*(105*A - x*(70*B + 3*x*(7*C + 4*D*x))) + d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + 33*a*b^2*d
*(1280*c^5*D - 128*c^4*d*(9*C - 5*D*x) + 16*c^3*d^2*(63*B - 2*x*(18*C + 5*D*x)) + 8*c^2*d^3*(-105*A + x*(63*B
+ 2*x*(9*C + 5*D*x))) + d^5*x^2*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^4*x*(210*A + x*(63*B + x*(36*C
+ 25*D*x)))) + b^3*(-15360*c^6*D + 1280*c^5*d*(11*C - 6*D*x) - 128*c^4*d^2*(99*B - 5*x*(11*C + 3*D*x)) + 16*c^
3*d^3*(693*A - 2*x*(198*B + 5*x*(11*C + 6*D*x))) + d^6*x^3*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) + 8*c^2*d
^4*x*(693*A + x*(198*B + 5*x*(22*C + 15*D*x))) - 2*c*d^5*x^2*(693*A + x*(396*B + 5*x*(55*C + 42*D*x))))))/(346
5*d^7*Sqrt[c + d*x])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1022\) vs. \(2(410)=820\).
time = 0.10, size = 1023, normalized size = 2.36 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/d^7*(1/11*D*b^3*(d*x+c)^(11/2)+3/7*C*a*b^2*d^2*(d*x+c)^(7/2)-5/7*C*b^3*c*d*(d*x+c)^(7/2)-A*b^3*c*d^3*(d*x+c)
^(3/2)+2*B*b^3*c^2*d^2*(d*x+c)^(3/2)-10/3*C*b^3*c^3*d*(d*x+c)^(3/2)-2*C*a^3*c*d^4*(d*x+c)^(1/2)+5*C*b^3*c^4*d*
(d*x+c)^(1/2)-4/5*B*b^3*c*d^2*(d*x+c)^(5/2)+3/5*C*a^2*b*d^3*(d*x+c)^(5/2)+2*C*b^3*c^2*d*(d*x+c)^(5/2)+3/5*B*a*
b^2*d^3*(d*x+c)^(5/2)+3/7*D*a^2*b*d^2*(d*x+c)^(7/2)-D*a^3*c*d^3*(d*x+c)^(3/2)+1/3*D*a*b^2*d*(d*x+c)^(9/2)+B*a^
2*b*d^4*(d*x+c)^(3/2)+A*a*b^2*d^4*(d*x+c)^(3/2)+3*D*a^3*c^2*d^3*(d*x+c)^(1/2)+3*A*a^2*b*d^5*(d*x+c)^(1/2)+3*A*
b^3*c^2*d^3*(d*x+c)^(1/2)-4*B*b^3*c^3*d^2*(d*x+c)^(1/2)+6*C*a*b^2*c^2*d^2*(d*x+c)^(3/2)-3*B*a*b^2*c*d^3*(d*x+c
)^(3/2)+6*D*a^2*b*c^2*d^2*(d*x+c)^(3/2)-10*D*a*b^2*c^3*d*(d*x+c)^(3/2)-12/5*C*a*b^2*c*d^2*(d*x+c)^(5/2)-12/5*D
*a^2*b*c*d^2*(d*x+c)^(5/2)+6*D*a*b^2*c^2*d*(d*x+c)^(5/2)-6*A*a*b^2*c*d^4*(d*x+c)^(1/2)-6*B*a^2*b*c*d^4*(d*x+c)
^(1/2)+9*B*a*b^2*c^2*d^3*(d*x+c)^(1/2)+9*C*a^2*b*c^2*d^3*(d*x+c)^(1/2)-3*C*a^2*b*c*d^3*(d*x+c)^(3/2)-15/7*D*a*
b^2*c*d*(d*x+c)^(7/2)+15*D*a*b^2*c^4*d*(d*x+c)^(1/2)+B*a^3*d^5*(d*x+c)^(1/2)+1/9*C*b^3*d*(d*x+c)^(9/2)-2/3*D*b
^3*c*(d*x+c)^(9/2)+1/7*B*b^3*d^2*(d*x+c)^(7/2)+15/7*D*b^3*c^2*(d*x+c)^(7/2)+1/5*D*a^3*d^3*(d*x+c)^(5/2)-4*D*b^
3*c^3*(d*x+c)^(5/2)+1/3*C*a^3*d^4*(d*x+c)^(3/2)+5*D*b^3*c^4*(d*x+c)^(3/2)-6*D*b^3*c^5*(d*x+c)^(1/2)-(A*a^3*d^6
-3*A*a^2*b*c*d^5+3*A*a*b^2*c^2*d^4-A*b^3*c^3*d^3-B*a^3*c*d^5+3*B*a^2*b*c^2*d^4-3*B*a*b^2*c^3*d^3+B*b^3*c^4*d^2
+C*a^3*c^2*d^4-3*C*a^2*b*c^3*d^3+3*C*a*b^2*c^4*d^2-C*b^3*c^5*d-D*a^3*c^3*d^3+3*D*a^2*b*c^4*d^2-3*D*a*b^2*c^5*d
+D*b^3*c^6)/(d*x+c)^(1/2)-12*D*a^2*b*c^3*d^2*(d*x+c)^(1/2)-12*C*a*b^2*c^3*d^2*(d*x+c)^(1/2)+1/5*A*b^3*d^3*(d*x
+c)^(5/2))

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 629, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (\frac {315 \, {\left (d x + c\right )}^{\frac {11}{2}} D b^{3} - 385 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 495 \, {\left (15 \, D b^{3} c^{2} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 693 \, {\left (20 \, D b^{3} c^{3} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (15 \, D b^{3} c^{4} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 3465 \, {\left (6 \, D b^{3} c^{5} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} \sqrt {d x + c}}{d^{6}} - \frac {3465 \, {\left (D b^{3} c^{6} + A a^{3} d^{6} - {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )}}{\sqrt {d x + c} d^{6}}\right )}}{3465 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/3465*((315*(d*x + c)^(11/2)*D*b^3 - 385*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d*x + c)^(9/2) + 495*(15*D*b^3*
c^2 - 5*(3*D*a*b^2 + C*b^3)*c*d + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^2)*(d*x + c)^(7/2) - 693*(20*D*b^3*c^3 - 1
0*(3*D*a*b^2 + C*b^3)*c^2*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3
)*d^3)*(d*x + c)^(5/2) + 1155*(15*D*b^3*c^4 - 10*(3*D*a*b^2 + C*b^3)*c^3*d + 6*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)
*c^2*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^4)*(d*x + c)^(3
/2) - 3465*(6*D*b^3*c^5 - 5*(3*D*a*b^2 + C*b^3)*c^4*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^2 - 3*(D*a^3 +
3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^4 - (B*a^3 + 3*A*a^2*b)*d^5)*s
qrt(d*x + c))/d^6 - 3465*(D*b^3*c^6 + A*a^3*d^6 - (3*D*a*b^2 + C*b^3)*c^5*d + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*
c^4*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 - (B*a^3 +
3*A*a^2*b)*c*d^5)/(sqrt(d*x + c)*d^6))/d

________________________________________________________________________________________

Fricas [A]
time = 1.18, size = 677, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (315 \, D b^{3} d^{6} x^{6} - 15360 \, D b^{3} c^{6} - 3465 \, A a^{3} d^{6} - 9240 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} + 6930 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 35 \, {\left (12 \, D b^{3} c d^{5} - 11 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 5 \, {\left (120 \, D b^{3} c^{2} d^{4} + 99 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6} - 110 \, {\left (3 \, D a b^{2} c + C b^{3} c\right )} d^{5}\right )} x^{4} + 11088 \, {\left (D a^{3} c^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3}\right )} d^{3} - {\left (960 \, D b^{3} c^{3} d^{3} - 693 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6} + 792 \, {\left (3 \, D a^{2} b c + {\left (3 \, C a b^{2} + B b^{3}\right )} c\right )} d^{5} - 880 \, {\left (3 \, D a b^{2} c^{2} + C b^{3} c^{2}\right )} d^{4}\right )} x^{3} - 12672 \, {\left (3 \, D a^{2} b c^{4} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4}\right )} d^{2} + {\left (1920 \, D b^{3} c^{4} d^{2} + 1155 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6} - 1386 \, {\left (D a^{3} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c\right )} d^{5} + 1584 \, {\left (3 \, D a^{2} b c^{2} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2}\right )} d^{4} - 1760 \, {\left (3 \, D a b^{2} c^{3} + C b^{3} c^{3}\right )} d^{3}\right )} x^{2} + 14080 \, {\left (3 \, D a b^{2} c^{5} + C b^{3} c^{5}\right )} d - {\left (7680 \, D b^{3} c^{5} d + 4620 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 3465 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6} - 5544 \, {\left (D a^{3} c^{2} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} d^{4} + 6336 \, {\left (3 \, D a^{2} b c^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3}\right )} d^{3} - 7040 \, {\left (3 \, D a b^{2} c^{4} + C b^{3} c^{4}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{3465 \, {\left (d^{8} x + c d^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/3465*(315*D*b^3*d^6*x^6 - 15360*D*b^3*c^6 - 3465*A*a^3*d^6 - 9240*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 +
6930*(B*a^3 + 3*A*a^2*b)*c*d^5 - 35*(12*D*b^3*c*d^5 - 11*(3*D*a*b^2 + C*b^3)*d^6)*x^5 + 5*(120*D*b^3*c^2*d^4 +
99*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6 - 110*(3*D*a*b^2*c + C*b^3*c)*d^5)*x^4 + 11088*(D*a^3*c^3 + (3*C*a^2*b
+ 3*B*a*b^2 + A*b^3)*c^3)*d^3 - (960*D*b^3*c^3*d^3 - 693*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^6 + 792*(3
*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^5 - 880*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^4)*x^3 - 12672*(3*D*a^2*b*c^4 + (3
*C*a*b^2 + B*b^3)*c^4)*d^2 + (1920*D*b^3*c^4*d^2 + 1155*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6 - 1386*(D*a^3*c +
(3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^5 + 1584*(3*D*a^2*b*c^2 + (3*C*a*b^2 + B*b^3)*c^2)*d^4 - 1760*(3*D*a*b^2*
c^3 + C*b^3*c^3)*d^3)*x^2 + 14080*(3*D*a*b^2*c^5 + C*b^3*c^5)*d - (7680*D*b^3*c^5*d + 4620*(C*a^3 + 3*B*a^2*b
+ 3*A*a*b^2)*c*d^5 - 3465*(B*a^3 + 3*A*a^2*b)*d^6 - 5544*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2)*d^4
+ 6336*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^3 - 7040*(3*D*a*b^2*c^4 + C*b^3*c^4)*d^2)*x)*sqrt(d*x + c)
/(d^8*x + c*d^7)

________________________________________________________________________________________

Sympy [A]
time = 92.94, size = 707, normalized size = 1.63 \begin {gather*} \frac {2 D b^{3} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{7}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (2 C b^{3} d + 6 D a b^{2} d - 12 D b^{3} c\right )}{9 d^{7}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 B b^{3} d^{2} + 6 C a b^{2} d^{2} - 10 C b^{3} c d + 6 D a^{2} b d^{2} - 30 D a b^{2} c d + 30 D b^{3} c^{2}\right )}{7 d^{7}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 A b^{3} d^{3} + 6 B a b^{2} d^{3} - 8 B b^{3} c d^{2} + 6 C a^{2} b d^{3} - 24 C a b^{2} c d^{2} + 20 C b^{3} c^{2} d + 2 D a^{3} d^{3} - 24 D a^{2} b c d^{2} + 60 D a b^{2} c^{2} d - 40 D b^{3} c^{3}\right )}{5 d^{7}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (6 A a b^{2} d^{4} - 6 A b^{3} c d^{3} + 6 B a^{2} b d^{4} - 18 B a b^{2} c d^{3} + 12 B b^{3} c^{2} d^{2} + 2 C a^{3} d^{4} - 18 C a^{2} b c d^{3} + 36 C a b^{2} c^{2} d^{2} - 20 C b^{3} c^{3} d - 6 D a^{3} c d^{3} + 36 D a^{2} b c^{2} d^{2} - 60 D a b^{2} c^{3} d + 30 D b^{3} c^{4}\right )}{3 d^{7}} + \frac {\sqrt {c + d x} \left (6 A a^{2} b d^{5} - 12 A a b^{2} c d^{4} + 6 A b^{3} c^{2} d^{3} + 2 B a^{3} d^{5} - 12 B a^{2} b c d^{4} + 18 B a b^{2} c^{2} d^{3} - 8 B b^{3} c^{3} d^{2} - 4 C a^{3} c d^{4} + 18 C a^{2} b c^{2} d^{3} - 24 C a b^{2} c^{3} d^{2} + 10 C b^{3} c^{4} d + 6 D a^{3} c^{2} d^{3} - 24 D a^{2} b c^{3} d^{2} + 30 D a b^{2} c^{4} d - 12 D b^{3} c^{5}\right )}{d^{7}} + \frac {2 \left (a d - b c\right )^{3} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{7} \sqrt {c + d x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)

[Out]

2*D*b**3*(c + d*x)**(11/2)/(11*d**7) + (c + d*x)**(9/2)*(2*C*b**3*d + 6*D*a*b**2*d - 12*D*b**3*c)/(9*d**7) + (
c + d*x)**(7/2)*(2*B*b**3*d**2 + 6*C*a*b**2*d**2 - 10*C*b**3*c*d + 6*D*a**2*b*d**2 - 30*D*a*b**2*c*d + 30*D*b*
*3*c**2)/(7*d**7) + (c + d*x)**(5/2)*(2*A*b**3*d**3 + 6*B*a*b**2*d**3 - 8*B*b**3*c*d**2 + 6*C*a**2*b*d**3 - 24
*C*a*b**2*c*d**2 + 20*C*b**3*c**2*d + 2*D*a**3*d**3 - 24*D*a**2*b*c*d**2 + 60*D*a*b**2*c**2*d - 40*D*b**3*c**3
)/(5*d**7) + (c + d*x)**(3/2)*(6*A*a*b**2*d**4 - 6*A*b**3*c*d**3 + 6*B*a**2*b*d**4 - 18*B*a*b**2*c*d**3 + 12*B
*b**3*c**2*d**2 + 2*C*a**3*d**4 - 18*C*a**2*b*c*d**3 + 36*C*a*b**2*c**2*d**2 - 20*C*b**3*c**3*d - 6*D*a**3*c*d
**3 + 36*D*a**2*b*c**2*d**2 - 60*D*a*b**2*c**3*d + 30*D*b**3*c**4)/(3*d**7) + sqrt(c + d*x)*(6*A*a**2*b*d**5 -
12*A*a*b**2*c*d**4 + 6*A*b**3*c**2*d**3 + 2*B*a**3*d**5 - 12*B*a**2*b*c*d**4 + 18*B*a*b**2*c**2*d**3 - 8*B*b*
*3*c**3*d**2 - 4*C*a**3*c*d**4 + 18*C*a**2*b*c**2*d**3 - 24*C*a*b**2*c**3*d**2 + 10*C*b**3*c**4*d + 6*D*a**3*c
**2*d**3 - 24*D*a**2*b*c**3*d**2 + 30*D*a*b**2*c**4*d - 12*D*b**3*c**5)/d**7 + 2*(a*d - b*c)**3*(-A*d**3 + B*c
*d**2 - C*c**2*d + D*c**3)/(d**7*sqrt(c + d*x))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (412) = 824\).
time = 0.68, size = 1067, normalized size = 2.46 \begin {gather*} -\frac {2 \, {\left (D b^{3} c^{6} - 3 \, D a b^{2} c^{5} d - C b^{3} c^{5} d + 3 \, D a^{2} b c^{4} d^{2} + 3 \, C a b^{2} c^{4} d^{2} + B b^{3} c^{4} d^{2} - D a^{3} c^{3} d^{3} - 3 \, C a^{2} b c^{3} d^{3} - 3 \, B a b^{2} c^{3} d^{3} - A b^{3} c^{3} d^{3} + C a^{3} c^{2} d^{4} + 3 \, B a^{2} b c^{2} d^{4} + 3 \, A a b^{2} c^{2} d^{4} - B a^{3} c d^{5} - 3 \, A a^{2} b c d^{5} + A a^{3} d^{6}\right )}}{\sqrt {d x + c} d^{7}} + \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} D b^{3} d^{70} - 2310 \, {\left (d x + c\right )}^{\frac {9}{2}} D b^{3} c d^{70} + 7425 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{3} c^{2} d^{70} - 13860 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{3} c^{3} d^{70} + 17325 \, {\left (d x + c\right )}^{\frac {3}{2}} D b^{3} c^{4} d^{70} - 20790 \, \sqrt {d x + c} D b^{3} c^{5} d^{70} + 1155 \, {\left (d x + c\right )}^{\frac {9}{2}} D a b^{2} d^{71} + 385 \, {\left (d x + c\right )}^{\frac {9}{2}} C b^{3} d^{71} - 7425 \, {\left (d x + c\right )}^{\frac {7}{2}} D a b^{2} c d^{71} - 2475 \, {\left (d x + c\right )}^{\frac {7}{2}} C b^{3} c d^{71} + 20790 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b^{2} c^{2} d^{71} + 6930 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{3} c^{2} d^{71} - 34650 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b^{2} c^{3} d^{71} - 11550 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{3} c^{3} d^{71} + 51975 \, \sqrt {d x + c} D a b^{2} c^{4} d^{71} + 17325 \, \sqrt {d x + c} C b^{3} c^{4} d^{71} + 1485 \, {\left (d x + c\right )}^{\frac {7}{2}} D a^{2} b d^{72} + 1485 \, {\left (d x + c\right )}^{\frac {7}{2}} C a b^{2} d^{72} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} B b^{3} d^{72} - 8316 \, {\left (d x + c\right )}^{\frac {5}{2}} D a^{2} b c d^{72} - 8316 \, {\left (d x + c\right )}^{\frac {5}{2}} C a b^{2} c d^{72} - 2772 \, {\left (d x + c\right )}^{\frac {5}{2}} B b^{3} c d^{72} + 20790 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} b c^{2} d^{72} + 20790 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b^{2} c^{2} d^{72} + 6930 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{3} c^{2} d^{72} - 41580 \, \sqrt {d x + c} D a^{2} b c^{3} d^{72} - 41580 \, \sqrt {d x + c} C a b^{2} c^{3} d^{72} - 13860 \, \sqrt {d x + c} B b^{3} c^{3} d^{72} + 693 \, {\left (d x + c\right )}^{\frac {5}{2}} D a^{3} d^{73} + 2079 \, {\left (d x + c\right )}^{\frac {5}{2}} C a^{2} b d^{73} + 2079 \, {\left (d x + c\right )}^{\frac {5}{2}} B a b^{2} d^{73} + 693 \, {\left (d x + c\right )}^{\frac {5}{2}} A b^{3} d^{73} - 3465 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{3} c d^{73} - 10395 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{2} b c d^{73} - 10395 \, {\left (d x + c\right )}^{\frac {3}{2}} B a b^{2} c d^{73} - 3465 \, {\left (d x + c\right )}^{\frac {3}{2}} A b^{3} c d^{73} + 10395 \, \sqrt {d x + c} D a^{3} c^{2} d^{73} + 31185 \, \sqrt {d x + c} C a^{2} b c^{2} d^{73} + 31185 \, \sqrt {d x + c} B a b^{2} c^{2} d^{73} + 10395 \, \sqrt {d x + c} A b^{3} c^{2} d^{73} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{3} d^{74} + 3465 \, {\left (d x + c\right )}^{\frac {3}{2}} B a^{2} b d^{74} + 3465 \, {\left (d x + c\right )}^{\frac {3}{2}} A a b^{2} d^{74} - 6930 \, \sqrt {d x + c} C a^{3} c d^{74} - 20790 \, \sqrt {d x + c} B a^{2} b c d^{74} - 20790 \, \sqrt {d x + c} A a b^{2} c d^{74} + 3465 \, \sqrt {d x + c} B a^{3} d^{75} + 10395 \, \sqrt {d x + c} A a^{2} b d^{75}\right )}}{3465 \, d^{77}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*(D*b^3*c^6 - 3*D*a*b^2*c^5*d - C*b^3*c^5*d + 3*D*a^2*b*c^4*d^2 + 3*C*a*b^2*c^4*d^2 + B*b^3*c^4*d^2 - D*a^3*
c^3*d^3 - 3*C*a^2*b*c^3*d^3 - 3*B*a*b^2*c^3*d^3 - A*b^3*c^3*d^3 + C*a^3*c^2*d^4 + 3*B*a^2*b*c^2*d^4 + 3*A*a*b^
2*c^2*d^4 - B*a^3*c*d^5 - 3*A*a^2*b*c*d^5 + A*a^3*d^6)/(sqrt(d*x + c)*d^7) + 2/3465*(315*(d*x + c)^(11/2)*D*b^
3*d^70 - 2310*(d*x + c)^(9/2)*D*b^3*c*d^70 + 7425*(d*x + c)^(7/2)*D*b^3*c^2*d^70 - 13860*(d*x + c)^(5/2)*D*b^3
*c^3*d^70 + 17325*(d*x + c)^(3/2)*D*b^3*c^4*d^70 - 20790*sqrt(d*x + c)*D*b^3*c^5*d^70 + 1155*(d*x + c)^(9/2)*D
*a*b^2*d^71 + 385*(d*x + c)^(9/2)*C*b^3*d^71 - 7425*(d*x + c)^(7/2)*D*a*b^2*c*d^71 - 2475*(d*x + c)^(7/2)*C*b^
3*c*d^71 + 20790*(d*x + c)^(5/2)*D*a*b^2*c^2*d^71 + 6930*(d*x + c)^(5/2)*C*b^3*c^2*d^71 - 34650*(d*x + c)^(3/2
)*D*a*b^2*c^3*d^71 - 11550*(d*x + c)^(3/2)*C*b^3*c^3*d^71 + 51975*sqrt(d*x + c)*D*a*b^2*c^4*d^71 + 17325*sqrt(
d*x + c)*C*b^3*c^4*d^71 + 1485*(d*x + c)^(7/2)*D*a^2*b*d^72 + 1485*(d*x + c)^(7/2)*C*a*b^2*d^72 + 495*(d*x + c
)^(7/2)*B*b^3*d^72 - 8316*(d*x + c)^(5/2)*D*a^2*b*c*d^72 - 8316*(d*x + c)^(5/2)*C*a*b^2*c*d^72 - 2772*(d*x + c
)^(5/2)*B*b^3*c*d^72 + 20790*(d*x + c)^(3/2)*D*a^2*b*c^2*d^72 + 20790*(d*x + c)^(3/2)*C*a*b^2*c^2*d^72 + 6930*
(d*x + c)^(3/2)*B*b^3*c^2*d^72 - 41580*sqrt(d*x + c)*D*a^2*b*c^3*d^72 - 41580*sqrt(d*x + c)*C*a*b^2*c^3*d^72 -
13860*sqrt(d*x + c)*B*b^3*c^3*d^72 + 693*(d*x + c)^(5/2)*D*a^3*d^73 + 2079*(d*x + c)^(5/2)*C*a^2*b*d^73 + 207
9*(d*x + c)^(5/2)*B*a*b^2*d^73 + 693*(d*x + c)^(5/2)*A*b^3*d^73 - 3465*(d*x + c)^(3/2)*D*a^3*c*d^73 - 10395*(d
*x + c)^(3/2)*C*a^2*b*c*d^73 - 10395*(d*x + c)^(3/2)*B*a*b^2*c*d^73 - 3465*(d*x + c)^(3/2)*A*b^3*c*d^73 + 1039
5*sqrt(d*x + c)*D*a^3*c^2*d^73 + 31185*sqrt(d*x + c)*C*a^2*b*c^2*d^73 + 31185*sqrt(d*x + c)*B*a*b^2*c^2*d^73 +
10395*sqrt(d*x + c)*A*b^3*c^2*d^73 + 1155*(d*x + c)^(3/2)*C*a^3*d^74 + 3465*(d*x + c)^(3/2)*B*a^2*b*d^74 + 34
65*(d*x + c)^(3/2)*A*a*b^2*d^74 - 6930*sqrt(d*x + c)*C*a^3*c*d^74 - 20790*sqrt(d*x + c)*B*a^2*b*c*d^74 - 20790
*sqrt(d*x + c)*A*a*b^2*c*d^74 + 3465*sqrt(d*x + c)*B*a^3*d^75 + 10395*sqrt(d*x + c)*A*a^2*b*d^75)/d^77

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2),x)

[Out]

int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2), x)

________________________________________________________________________________________