3.18 \(\int \frac{(a+b x)^3 (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.352548, antiderivative size = 434, normalized size of antiderivative = 1., 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} \[ \frac{2 (c+d x)^{3/2} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{d^7}+\frac{2 b (c+d x)^{5/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{5 d^7}+\frac{2 (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{d^7 \sqrt{c+d x}}+\frac{2 (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^7 (c+d x)^{3/2}}+\frac{2 b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x)^(3/2)) + (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)))/(d^7*Sqrt[c + d*x]) - (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)
)*Sqrt[c + d*x])/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)^(3/2))/(3*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)^(5/2))/(5*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d
*x)^(7/2))/(7*d^7) + (2*b^3*D*(c + d*x)^(9/2))/(9*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)^{5/2}} \, 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)^{5/2}}+\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 (c+d x)^{3/2}}+\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 )}{d^6 \sqrt{c+d x}}+\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 ) \sqrt{c+d x}}{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)^{3/2}}{d^6}+\frac{b^2 (b C d-6 b c D+3 a d D) (c+d x)^{5/2}}{d^6}+\frac{b^3 D (c+d x)^{7/2}}{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 )}{3 d^7 (c+d x)^{3/2}}+\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 )}{d^7 \sqrt{c+d x}}-\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 ) \sqrt{c+d x}}{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)^{3/2}}{3 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)^{5/2}}{5 d^7}+\frac{2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{7/2}}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7}\\ \end{align*}

Mathematica [A]  time = 0.715992, size = 391, normalized size = 0.9 \[ \frac{2 \left (105 (c+d x)^3 \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )-315 (c+d x)^2 (b c-a d) \left (a^2 d^2 (C d-3 c D)+a b d \left (3 B d^2+15 c^2 D-8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )+63 b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )+315 (c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2-5 c^2 C d+6 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+105 (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )+45 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+35 b^3 D (c+d x)^6\right )}{315 d^7 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(105*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) + 315*(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) - 315*(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 + 105*(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 + 63*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 + 45*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 35*b^3*D*(c + d*x)^6))/(315*d^7*(c + d*x)^(3/
2))

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Maple [B]  time = 0.008, size = 841, normalized size = 1.9 \begin{align*} -{\frac{-70\,{b}^{3}D{x}^{6}{d}^{6}-90\,C{b}^{3}{d}^{6}{x}^{5}-270\,Da{b}^{2}{d}^{6}{x}^{5}+120\,D{b}^{3}c{d}^{5}{x}^{5}-126\,B{b}^{3}{d}^{6}{x}^{4}-378\,Ca{b}^{2}{d}^{6}{x}^{4}+180\,C{b}^{3}c{d}^{5}{x}^{4}-378\,D{a}^{2}b{d}^{6}{x}^{4}+540\,Da{b}^{2}c{d}^{5}{x}^{4}-240\,D{b}^{3}{c}^{2}{d}^{4}{x}^{4}-210\,A{b}^{3}{d}^{6}{x}^{3}-630\,Ba{b}^{2}{d}^{6}{x}^{3}+336\,B{b}^{3}c{d}^{5}{x}^{3}-630\,C{a}^{2}b{d}^{6}{x}^{3}+1008\,Ca{b}^{2}c{d}^{5}{x}^{3}-480\,C{b}^{3}{c}^{2}{d}^{4}{x}^{3}-210\,D{a}^{3}{d}^{6}{x}^{3}+1008\,D{a}^{2}bc{d}^{5}{x}^{3}-1440\,Da{b}^{2}{c}^{2}{d}^{4}{x}^{3}+640\,D{b}^{3}{c}^{3}{d}^{3}{x}^{3}-1890\,Aa{b}^{2}{d}^{6}{x}^{2}+1260\,A{b}^{3}c{d}^{5}{x}^{2}-1890\,B{a}^{2}b{d}^{6}{x}^{2}+3780\,Ba{b}^{2}c{d}^{5}{x}^{2}-2016\,B{b}^{3}{c}^{2}{d}^{4}{x}^{2}-630\,C{a}^{3}{d}^{6}{x}^{2}+3780\,C{a}^{2}bc{d}^{5}{x}^{2}-6048\,Ca{b}^{2}{c}^{2}{d}^{4}{x}^{2}+2880\,C{b}^{3}{c}^{3}{d}^{3}{x}^{2}+1260\,D{a}^{3}c{d}^{5}{x}^{2}-6048\,D{a}^{2}b{c}^{2}{d}^{4}{x}^{2}+8640\,Da{b}^{2}{c}^{3}{d}^{3}{x}^{2}-3840\,D{b}^{3}{c}^{4}{d}^{2}{x}^{2}+1890\,A{a}^{2}b{d}^{6}x-7560\,Aa{b}^{2}c{d}^{5}x+5040\,A{b}^{3}{c}^{2}{d}^{4}x+630\,B{a}^{3}{d}^{6}x-7560\,B{a}^{2}bc{d}^{5}x+15120\,Ba{b}^{2}{c}^{2}{d}^{4}x-8064\,B{b}^{3}{c}^{3}{d}^{3}x-2520\,C{a}^{3}c{d}^{5}x+15120\,C{a}^{2}b{c}^{2}{d}^{4}x-24192\,Ca{b}^{2}{c}^{3}{d}^{3}x+11520\,C{b}^{3}{c}^{4}{d}^{2}x+5040\,D{a}^{3}{c}^{2}{d}^{4}x-24192\,D{a}^{2}b{c}^{3}{d}^{3}x+34560\,Da{b}^{2}{c}^{4}{d}^{2}x-15360\,D{b}^{3}{c}^{5}dx+210\,{a}^{3}A{d}^{6}+1260\,A{a}^{2}bc{d}^{5}-5040\,Aa{b}^{2}{c}^{2}{d}^{4}+3360\,A{b}^{3}{c}^{3}{d}^{3}+420\,B{a}^{3}c{d}^{5}-5040\,B{a}^{2}b{c}^{2}{d}^{4}+10080\,Ba{b}^{2}{c}^{3}{d}^{3}-5376\,B{b}^{3}{c}^{4}{d}^{2}-1680\,C{a}^{3}{c}^{2}{d}^{4}+10080\,C{a}^{2}b{c}^{3}{d}^{3}-16128\,Ca{b}^{2}{c}^{4}{d}^{2}+7680\,C{b}^{3}{c}^{5}d+3360\,D{a}^{3}{c}^{3}{d}^{3}-16128\,D{a}^{2}b{c}^{4}{d}^{2}+23040\,Da{b}^{2}{c}^{5}d-10240\,D{b}^{3}{c}^{6}}{315\,{d}^{7}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

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)^(5/2),x)

[Out]

-2/315/(d*x+c)^(3/2)*(-35*D*b^3*d^6*x^6-45*C*b^3*d^6*x^5-135*D*a*b^2*d^6*x^5+60*D*b^3*c*d^5*x^5-63*B*b^3*d^6*x
^4-189*C*a*b^2*d^6*x^4+90*C*b^3*c*d^5*x^4-189*D*a^2*b*d^6*x^4+270*D*a*b^2*c*d^5*x^4-120*D*b^3*c^2*d^4*x^4-105*
A*b^3*d^6*x^3-315*B*a*b^2*d^6*x^3+168*B*b^3*c*d^5*x^3-315*C*a^2*b*d^6*x^3+504*C*a*b^2*c*d^5*x^3-240*C*b^3*c^2*
d^4*x^3-105*D*a^3*d^6*x^3+504*D*a^2*b*c*d^5*x^3-720*D*a*b^2*c^2*d^4*x^3+320*D*b^3*c^3*d^3*x^3-945*A*a*b^2*d^6*
x^2+630*A*b^3*c*d^5*x^2-945*B*a^2*b*d^6*x^2+1890*B*a*b^2*c*d^5*x^2-1008*B*b^3*c^2*d^4*x^2-315*C*a^3*d^6*x^2+18
90*C*a^2*b*c*d^5*x^2-3024*C*a*b^2*c^2*d^4*x^2+1440*C*b^3*c^3*d^3*x^2+630*D*a^3*c*d^5*x^2-3024*D*a^2*b*c^2*d^4*
x^2+4320*D*a*b^2*c^3*d^3*x^2-1920*D*b^3*c^4*d^2*x^2+945*A*a^2*b*d^6*x-3780*A*a*b^2*c*d^5*x+2520*A*b^3*c^2*d^4*
x+315*B*a^3*d^6*x-3780*B*a^2*b*c*d^5*x+7560*B*a*b^2*c^2*d^4*x-4032*B*b^3*c^3*d^3*x-1260*C*a^3*c*d^5*x+7560*C*a
^2*b*c^2*d^4*x-12096*C*a*b^2*c^3*d^3*x+5760*C*b^3*c^4*d^2*x+2520*D*a^3*c^2*d^4*x-12096*D*a^2*b*c^3*d^3*x+17280
*D*a*b^2*c^4*d^2*x-7680*D*b^3*c^5*d*x+105*A*a^3*d^6+630*A*a^2*b*c*d^5-2520*A*a*b^2*c^2*d^4+1680*A*b^3*c^3*d^3+
210*B*a^3*c*d^5-2520*B*a^2*b*c^2*d^4+5040*B*a*b^2*c^3*d^3-2688*B*b^3*c^4*d^2-840*C*a^3*c^2*d^4+5040*C*a^2*b*c^
3*d^3-8064*C*a*b^2*c^4*d^2+3840*C*b^3*c^5*d+1680*D*a^3*c^3*d^3-8064*D*a^2*b*c^4*d^2+11520*D*a*b^2*c^5*d-5120*D
*b^3*c^6)/d^7

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Maxima [A]  time = 1.64268, size = 846, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b^{3} - 45 \,{\left (6 \, D b^{3} c -{\left (3 \, D a b^{2} + C b^{3}\right )} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\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{5}{2}} - 105 \,{\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{3}{2}} + 315 \,{\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 )} \sqrt{d x + c}}{d^{6}} - \frac{105 \,{\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} - 3 \,{\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 )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{6}}\right )}}{315 \, d} \end{align*}

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)^(5/2),x, algorithm="maxima")

[Out]

2/315*((35*(d*x + c)^(9/2)*D*b^3 - 45*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d*x + c)^(7/2) + 63*(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)^(5/2) - 105*(20*D*b^3*c^3 - 10*(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)^(3/2) + 315*(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)*sqrt(d*x + c))/d^6
- 105*(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
- 3*(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)*(d*x + c
))/((d*x + c)^(3/2)*d^6))/d

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.40143, size = 1391, normalized size = 3.21 \begin{align*} \text{result too large to display} \end{align*}

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)^(5/2),x, algorithm="giac")

[Out]

2/3*(18*(d*x + c)*D*b^3*c^5 - D*b^3*c^6 - 45*(d*x + c)*D*a*b^2*c^4*d - 15*(d*x + c)*C*b^3*c^4*d + 3*D*a*b^2*c^
5*d + C*b^3*c^5*d + 36*(d*x + c)*D*a^2*b*c^3*d^2 + 36*(d*x + c)*C*a*b^2*c^3*d^2 + 12*(d*x + c)*B*b^3*c^3*d^2 -
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 - 9*(d*x + c)*D*a^3*c^2*d^3 - 27*(d*x + c)*C*a^2*b*c^2*
d^3 - 27*(d*x + c)*B*a*b^2*c^2*d^3 - 9*(d*x + c)*A*b^3*c^2*d^3 + 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 + 6*(d*x + c)*C*a^3*c*d^4 + 18*(d*x + c)*B*a^2*b*c*d^4 + 18*(d*x + c)*A*a*b^2*c*d^4 -
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 - 3*(d*x + c)*B*a^3*d^5 - 9*(d*x + c)*A*a^2*b*d^5 + B*a
^3*c*d^5 + 3*A*a^2*b*c*d^5 - A*a^3*d^6)/((d*x + c)^(3/2)*d^7) + 2/315*(35*(d*x + c)^(9/2)*D*b^3*d^56 - 270*(d*
x + c)^(7/2)*D*b^3*c*d^56 + 945*(d*x + c)^(5/2)*D*b^3*c^2*d^56 - 2100*(d*x + c)^(3/2)*D*b^3*c^3*d^56 + 4725*sq
rt(d*x + c)*D*b^3*c^4*d^56 + 135*(d*x + c)^(7/2)*D*a*b^2*d^57 + 45*(d*x + c)^(7/2)*C*b^3*d^57 - 945*(d*x + c)^
(5/2)*D*a*b^2*c*d^57 - 315*(d*x + c)^(5/2)*C*b^3*c*d^57 + 3150*(d*x + c)^(3/2)*D*a*b^2*c^2*d^57 + 1050*(d*x +
c)^(3/2)*C*b^3*c^2*d^57 - 9450*sqrt(d*x + c)*D*a*b^2*c^3*d^57 - 3150*sqrt(d*x + c)*C*b^3*c^3*d^57 + 189*(d*x +
c)^(5/2)*D*a^2*b*d^58 + 189*(d*x + c)^(5/2)*C*a*b^2*d^58 + 63*(d*x + c)^(5/2)*B*b^3*d^58 - 1260*(d*x + c)^(3/
2)*D*a^2*b*c*d^58 - 1260*(d*x + c)^(3/2)*C*a*b^2*c*d^58 - 420*(d*x + c)^(3/2)*B*b^3*c*d^58 + 5670*sqrt(d*x + c
)*D*a^2*b*c^2*d^58 + 5670*sqrt(d*x + c)*C*a*b^2*c^2*d^58 + 1890*sqrt(d*x + c)*B*b^3*c^2*d^58 + 105*(d*x + c)^(
3/2)*D*a^3*d^59 + 315*(d*x + c)^(3/2)*C*a^2*b*d^59 + 315*(d*x + c)^(3/2)*B*a*b^2*d^59 + 105*(d*x + c)^(3/2)*A*
b^3*d^59 - 945*sqrt(d*x + c)*D*a^3*c*d^59 - 2835*sqrt(d*x + c)*C*a^2*b*c*d^59 - 2835*sqrt(d*x + c)*B*a*b^2*c*d
^59 - 945*sqrt(d*x + c)*A*b^3*c*d^59 + 315*sqrt(d*x + c)*C*a^3*d^60 + 945*sqrt(d*x + c)*B*a^2*b*d^60 + 945*sqr
t(d*x + c)*A*a*b^2*d^60)/d^63