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

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

[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

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Rubi [A]  time = 0.36, 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 = {1620} \[ \frac {2 (c+d x)^{5/2} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{5 d^7}-\frac {2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{3 d^7}+\frac {2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{d^7}+\frac {2 (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \]

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 1620

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*}

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Mathematica [A]  time = 1.16, size = 391, normalized size = 0.90 \[ \frac {2 \left (-1155 (c+d x)^2 (b c-a d) \left (a^2 d^2 (C d-3 c D)+a b d \left (3 B d^2+15 c^2 D-8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+495 b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )+693 (c+d x)^3 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )-3465 (c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+3465 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+385 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+315 b^3 D (c+d x)^6\right )}{3465 d^7 \sqrt {c+d x}} \]

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*(3465*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) - 3465*(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))*(c + d*x) - 1155*(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)^2 + 693*
(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)^3 + 495*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)^4 + 385*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 315*b^3*D*(c + d*x)^6))/(3465*d^7*Sqrt[c
+ d*x])

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fricas [A]  time = 1.22, size = 677, normalized size = 1.56 \[ \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 )}} \]

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)

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giac [B]  time = 1.49, size = 1067, normalized size = 2.46 \[ \text {result too large to display} \]

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

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maple [B]  time = 0.01, size = 841, normalized size = 1.94 \[ -\frac {2 \left (-315 b^{3} D x^{6} d^{6}-385 C \,b^{3} d^{6} x^{5}-1155 D a \,b^{2} d^{6} x^{5}+420 D b^{3} c \,d^{5} x^{5}-495 B \,b^{3} d^{6} x^{4}-1485 C a \,b^{2} d^{6} x^{4}+550 C \,b^{3} c \,d^{5} x^{4}-1485 D a^{2} b \,d^{6} x^{4}+1650 D a \,b^{2} c \,d^{5} x^{4}-600 D b^{3} c^{2} d^{4} x^{4}-693 A \,b^{3} d^{6} x^{3}-2079 B a \,b^{2} d^{6} x^{3}+792 B \,b^{3} c \,d^{5} x^{3}-2079 C \,a^{2} b \,d^{6} x^{3}+2376 C a \,b^{2} c \,d^{5} x^{3}-880 C \,b^{3} c^{2} d^{4} x^{3}-693 D a^{3} d^{6} x^{3}+2376 D a^{2} b c \,d^{5} x^{3}-2640 D a \,b^{2} c^{2} d^{4} x^{3}+960 D b^{3} c^{3} d^{3} x^{3}-3465 A a \,b^{2} d^{6} x^{2}+1386 A \,b^{3} c \,d^{5} x^{2}-3465 B \,a^{2} b \,d^{6} x^{2}+4158 B a \,b^{2} c \,d^{5} x^{2}-1584 B \,b^{3} c^{2} d^{4} x^{2}-1155 C \,a^{3} d^{6} x^{2}+4158 C \,a^{2} b c \,d^{5} x^{2}-4752 C a \,b^{2} c^{2} d^{4} x^{2}+1760 C \,b^{3} c^{3} d^{3} x^{2}+1386 D a^{3} c \,d^{5} x^{2}-4752 D a^{2} b \,c^{2} d^{4} x^{2}+5280 D a \,b^{2} c^{3} d^{3} x^{2}-1920 D b^{3} c^{4} d^{2} x^{2}-10395 A \,a^{2} b \,d^{6} x +13860 A a \,b^{2} c \,d^{5} x -5544 A \,b^{3} c^{2} d^{4} x -3465 B \,a^{3} d^{6} x +13860 B \,a^{2} b c \,d^{5} x -16632 B a \,b^{2} c^{2} d^{4} x +6336 B \,b^{3} c^{3} d^{3} x +4620 C \,a^{3} c \,d^{5} x -16632 C \,a^{2} b \,c^{2} d^{4} x +19008 C a \,b^{2} c^{3} d^{3} x -7040 C \,b^{3} c^{4} d^{2} x -5544 D a^{3} c^{2} d^{4} x +19008 D a^{2} b \,c^{3} d^{3} x -21120 D a \,b^{2} c^{4} d^{2} x +7680 D b^{3} c^{5} d x +3465 a^{3} A \,d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {d x +c}\, d^{7}} \]

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)

[Out]

-2/3465/(d*x+c)^(1/2)*(-315*D*b^3*d^6*x^6-385*C*b^3*d^6*x^5-1155*D*a*b^2*d^6*x^5+420*D*b^3*c*d^5*x^5-495*B*b^3
*d^6*x^4-1485*C*a*b^2*d^6*x^4+550*C*b^3*c*d^5*x^4-1485*D*a^2*b*d^6*x^4+1650*D*a*b^2*c*d^5*x^4-600*D*b^3*c^2*d^
4*x^4-693*A*b^3*d^6*x^3-2079*B*a*b^2*d^6*x^3+792*B*b^3*c*d^5*x^3-2079*C*a^2*b*d^6*x^3+2376*C*a*b^2*c*d^5*x^3-8
80*C*b^3*c^2*d^4*x^3-693*D*a^3*d^6*x^3+2376*D*a^2*b*c*d^5*x^3-2640*D*a*b^2*c^2*d^4*x^3+960*D*b^3*c^3*d^3*x^3-3
465*A*a*b^2*d^6*x^2+1386*A*b^3*c*d^5*x^2-3465*B*a^2*b*d^6*x^2+4158*B*a*b^2*c*d^5*x^2-1584*B*b^3*c^2*d^4*x^2-11
55*C*a^3*d^6*x^2+4158*C*a^2*b*c*d^5*x^2-4752*C*a*b^2*c^2*d^4*x^2+1760*C*b^3*c^3*d^3*x^2+1386*D*a^3*c*d^5*x^2-4
752*D*a^2*b*c^2*d^4*x^2+5280*D*a*b^2*c^3*d^3*x^2-1920*D*b^3*c^4*d^2*x^2-10395*A*a^2*b*d^6*x+13860*A*a*b^2*c*d^
5*x-5544*A*b^3*c^2*d^4*x-3465*B*a^3*d^6*x+13860*B*a^2*b*c*d^5*x-16632*B*a*b^2*c^2*d^4*x+6336*B*b^3*c^3*d^3*x+4
620*C*a^3*c*d^5*x-16632*C*a^2*b*c^2*d^4*x+19008*C*a*b^2*c^3*d^3*x-7040*C*b^3*c^4*d^2*x-5544*D*a^3*c^2*d^4*x+19
008*D*a^2*b*c^3*d^3*x-21120*D*a*b^2*c^4*d^2*x+7680*D*b^3*c^5*d*x+3465*A*a^3*d^6-20790*A*a^2*b*c*d^5+27720*A*a*
b^2*c^2*d^4-11088*A*b^3*c^3*d^3-6930*B*a^3*c*d^5+27720*B*a^2*b*c^2*d^4-33264*B*a*b^2*c^3*d^3+12672*B*b^3*c^4*d
^2+9240*C*a^3*c^2*d^4-33264*C*a^2*b*c^3*d^3+38016*C*a*b^2*c^4*d^2-14080*C*b^3*c^5*d-11088*D*a^3*c^3*d^3+38016*
D*a^2*b*c^4*d^2-42240*D*a*b^2*c^5*d+15360*D*b^3*c^6)/d^7

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maxima [A]  time = 0.47, size = 629, normalized size = 1.45 \[ \frac {2 \, {\left (\frac {315 \, {\left (d x + c\right )}^{\frac {11}{2}} D b^{3} - 385 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 495 \, {\left (15 \, D b^{3} c^{2} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 693 \, {\left (20 \, D b^{3} c^{3} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (15 \, D b^{3} c^{4} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 3465 \, {\left (6 \, D b^{3} c^{5} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} \sqrt {d x + c}}{d^{6}} - \frac {3465 \, {\left (D b^{3} c^{6} + A a^{3} d^{6} - {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )}}{\sqrt {d x + c} d^{6}}\right )}}{3465 \, d} \]

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

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

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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