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

Optimal. Leaf size=436 \[ -\frac {2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{5 d^7}+\frac {2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}+\frac {2 (c+d x)^{7/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{7 d^7}-\frac {2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{3 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7}+\frac {2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac {2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

[Out]

-2/3*(-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)^(3/2)/d^7-2/5
*(-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)^(5/2)/d^7+2/7*(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)^(7/2)/d^7+2/9*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-15*D*
c^2))*(d*x+c)^(9/2)/d^7+2/11*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(11/2)/d^7+2/13*b^3*D*(d*x+c)^(13/2)/d^7-2*(-
a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/2)/d^7

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Rubi [A]  time = 0.41, antiderivative size = 436, 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} \[ \frac {2 (c+d x)^{7/2} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{7 d^7}-\frac {2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{5 d^7}+\frac {2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}-\frac {2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{3 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7}+\frac {2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac {2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (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))*(c + d*x)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(
c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*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 )}{\sqrt {c+d x}} \, 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 \sqrt {c+d x}}+\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 ) \sqrt {c+d x}}{d^6}+\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 ) (c+d x)^{3/2}}{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)^{5/2}}{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)^{7/2}}{d^6}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{d^6}+\frac {b^3 D (c+d x)^{11/2}}{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 ) \sqrt {c+d x}}{d^7}-\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 ) (c+d x)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 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)^{9/2}}{9 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{11/2}}{11 d^7}+\frac {2 b^3 D (c+d x)^{13/2}}{13 d^7}\\ \end {align*}

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Mathematica [A]  time = 1.31, size = 391, normalized size = 0.90 \[ \frac {2 \sqrt {c+d x} \left (-9009 (c+d x)^2 (b c-a d) \left (a^2 d^2 (C d-3 c D)+a b d \left (3 B d^2+15 c^2 D-8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+5005 b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )+6435 (c+d x)^3 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )-15015 (c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+45045 (b c-a d)^3 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )+4095 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+3465 b^3 D (c+d x)^6\right )}{45045 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(45045*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) - 15015*(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) - 9009*(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 + 6435*(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 + 5005*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 + 4095*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 3465*b^3*D*(c + d*
x)^6))/(45045*d^7)

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fricas [A]  time = 0.76, size = 668, normalized size = 1.53 \[ \frac {2 \, {\left (3465 \, D b^{3} d^{6} x^{6} + 15360 \, D b^{3} c^{6} + 45045 \, A a^{3} d^{6} + 24024 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - 30030 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 315 \, {\left (12 \, D b^{3} c d^{5} - 13 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 35 \, {\left (120 \, D b^{3} c^{2} d^{4} + 143 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6} - 130 \, {\left (3 \, D a b^{2} c + C b^{3} c\right )} d^{5}\right )} x^{4} - 20592 \, {\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} - 5 \, {\left (960 \, D b^{3} c^{3} d^{3} - 1287 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6} + 1144 \, {\left (3 \, D a^{2} b c + {\left (3 \, C a b^{2} + B b^{3}\right )} c\right )} d^{5} - 1040 \, {\left (3 \, D a b^{2} c^{2} + C b^{3} c^{2}\right )} d^{4}\right )} x^{3} + 18304 \, {\left (3 \, D a^{2} b c^{4} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4}\right )} d^{2} + 3 \, {\left (1920 \, D b^{3} c^{4} d^{2} + 3003 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6} - 2574 \, {\left (D a^{3} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c\right )} d^{5} + 2288 \, {\left (3 \, D a^{2} b c^{2} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2}\right )} d^{4} - 2080 \, {\left (3 \, D a b^{2} c^{3} + C b^{3} c^{3}\right )} d^{3}\right )} x^{2} - 16640 \, {\left (3 \, D a b^{2} c^{5} + C b^{3} c^{5}\right )} d - {\left (7680 \, D b^{3} c^{5} d + 12012 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 15015 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6} - 10296 \, {\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} + 9152 \, {\left (3 \, D a^{2} b c^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3}\right )} d^{3} - 8320 \, {\left (3 \, D a b^{2} c^{4} + C b^{3} c^{4}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{7}} \]

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)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*D*b^3*d^6*x^6 + 15360*D*b^3*c^6 + 45045*A*a^3*d^6 + 24024*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^
4 - 30030*(B*a^3 + 3*A*a^2*b)*c*d^5 - 315*(12*D*b^3*c*d^5 - 13*(3*D*a*b^2 + C*b^3)*d^6)*x^5 + 35*(120*D*b^3*c^
2*d^4 + 143*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6 - 130*(3*D*a*b^2*c + C*b^3*c)*d^5)*x^4 - 20592*(D*a^3*c^3 + (3
*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3)*d^3 - 5*(960*D*b^3*c^3*d^3 - 1287*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d
^6 + 1144*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^5 - 1040*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^4)*x^3 + 18304*(3*D*a
^2*b*c^4 + (3*C*a*b^2 + B*b^3)*c^4)*d^2 + 3*(1920*D*b^3*c^4*d^2 + 3003*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6 - 2
574*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^5 + 2288*(3*D*a^2*b*c^2 + (3*C*a*b^2 + B*b^3)*c^2)*d^4 - 2
080*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^3)*x^2 - 16640*(3*D*a*b^2*c^5 + C*b^3*c^5)*d - (7680*D*b^3*c^5*d + 12012*(C*
a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 15015*(B*a^3 + 3*A*a^2*b)*d^6 - 10296*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^2
+ A*b^3)*c^2)*d^4 + 9152*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^3 - 8320*(3*D*a*b^2*c^4 + C*b^3*c^4)*d^2
)*x)*sqrt(d*x + c)/d^7

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giac [B]  time = 1.30, size = 854, normalized size = 1.96 \[ \text {result too large to display} \]

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)^(1/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(d*x + c)*A*a^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^3/d + 45045*((d*x + c)^(3
/2) - 3*sqrt(d*x + c)*c)*A*a^2*b/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C*
a^3/d^2 + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*B*a^2*b/d^2 + 9009*(3*(d*x +
c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*A*a*b^2/d^2 + 1287*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(
5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*D*a^3/d^3 + 3861*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*C*a^2*b/d^3 + 3861*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*B*a*b^2/d^3 + 1287*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*A*b^3/d^3 + 429*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2
)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*D*a^2*b/d^4 + 429*(35*(d*x +
c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*
C*a*b^2/d^4 + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*
c^3 + 315*sqrt(d*x + c)*c^4)*B*b^3/d^4 + 195*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2
)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*D*a*b^2/d^5 + 65*(63*(d*x
+ c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/
2)*c^4 - 693*sqrt(d*x + c)*c^5)*C*b^3/d^5 + 15*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c
)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x +
c)*c^6)*D*b^3/d^6)/d

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maple [B]  time = 0.01, size = 841, normalized size = 1.93 \[ \frac {2 \sqrt {d x +c}\, \left (3465 b^{3} D x^{6} d^{6}+4095 C \,b^{3} d^{6} x^{5}+12285 D a \,b^{2} d^{6} x^{5}-3780 D b^{3} c \,d^{5} x^{5}+5005 B \,b^{3} d^{6} x^{4}+15015 C a \,b^{2} d^{6} x^{4}-4550 C \,b^{3} c \,d^{5} x^{4}+15015 D a^{2} b \,d^{6} x^{4}-13650 D a \,b^{2} c \,d^{5} x^{4}+4200 D b^{3} c^{2} d^{4} x^{4}+6435 A \,b^{3} d^{6} x^{3}+19305 B a \,b^{2} d^{6} x^{3}-5720 B \,b^{3} c \,d^{5} x^{3}+19305 C \,a^{2} b \,d^{6} x^{3}-17160 C a \,b^{2} c \,d^{5} x^{3}+5200 C \,b^{3} c^{2} d^{4} x^{3}+6435 D a^{3} d^{6} x^{3}-17160 D a^{2} b c \,d^{5} x^{3}+15600 D a \,b^{2} c^{2} d^{4} x^{3}-4800 D b^{3} c^{3} d^{3} x^{3}+27027 A a \,b^{2} d^{6} x^{2}-7722 A \,b^{3} c \,d^{5} x^{2}+27027 B \,a^{2} b \,d^{6} x^{2}-23166 B a \,b^{2} c \,d^{5} x^{2}+6864 B \,b^{3} c^{2} d^{4} x^{2}+9009 C \,a^{3} d^{6} x^{2}-23166 C \,a^{2} b c \,d^{5} x^{2}+20592 C a \,b^{2} c^{2} d^{4} x^{2}-6240 C \,b^{3} c^{3} d^{3} x^{2}-7722 D a^{3} c \,d^{5} x^{2}+20592 D a^{2} b \,c^{2} d^{4} x^{2}-18720 D a \,b^{2} c^{3} d^{3} x^{2}+5760 D b^{3} c^{4} d^{2} x^{2}+45045 A \,a^{2} b \,d^{6} x -36036 A a \,b^{2} c \,d^{5} x +10296 A \,b^{3} c^{2} d^{4} x +15015 B \,a^{3} d^{6} x -36036 B \,a^{2} b c \,d^{5} x +30888 B a \,b^{2} c^{2} d^{4} x -9152 B \,b^{3} c^{3} d^{3} x -12012 C \,a^{3} c \,d^{5} x +30888 C \,a^{2} b \,c^{2} d^{4} x -27456 C a \,b^{2} c^{3} d^{3} x +8320 C \,b^{3} c^{4} d^{2} x +10296 D a^{3} c^{2} d^{4} x -27456 D a^{2} b \,c^{3} d^{3} x +24960 D a \,b^{2} c^{4} d^{2} x -7680 D b^{3} c^{5} d x +45045 a^{3} A \,d^{6}-90090 A \,a^{2} b c \,d^{5}+72072 A a \,b^{2} c^{2} d^{4}-20592 A \,b^{3} c^{3} d^{3}-30030 B \,a^{3} c \,d^{5}+72072 B \,a^{2} b \,c^{2} d^{4}-61776 B a \,b^{2} c^{3} d^{3}+18304 B \,b^{3} c^{4} d^{2}+24024 C \,a^{3} c^{2} d^{4}-61776 C \,a^{2} b \,c^{3} d^{3}+54912 C a \,b^{2} c^{4} d^{2}-16640 C \,b^{3} c^{5} d -20592 D a^{3} c^{3} d^{3}+54912 D a^{2} b \,c^{4} d^{2}-49920 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{45045 d^{7}} \]

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)^(1/2),x)

[Out]

2/45045*(d*x+c)^(1/2)*(3465*D*b^3*d^6*x^6+4095*C*b^3*d^6*x^5+12285*D*a*b^2*d^6*x^5-3780*D*b^3*c*d^5*x^5+5005*B
*b^3*d^6*x^4+15015*C*a*b^2*d^6*x^4-4550*C*b^3*c*d^5*x^4+15015*D*a^2*b*d^6*x^4-13650*D*a*b^2*c*d^5*x^4+4200*D*b
^3*c^2*d^4*x^4+6435*A*b^3*d^6*x^3+19305*B*a*b^2*d^6*x^3-5720*B*b^3*c*d^5*x^3+19305*C*a^2*b*d^6*x^3-17160*C*a*b
^2*c*d^5*x^3+5200*C*b^3*c^2*d^4*x^3+6435*D*a^3*d^6*x^3-17160*D*a^2*b*c*d^5*x^3+15600*D*a*b^2*c^2*d^4*x^3-4800*
D*b^3*c^3*d^3*x^3+27027*A*a*b^2*d^6*x^2-7722*A*b^3*c*d^5*x^2+27027*B*a^2*b*d^6*x^2-23166*B*a*b^2*c*d^5*x^2+686
4*B*b^3*c^2*d^4*x^2+9009*C*a^3*d^6*x^2-23166*C*a^2*b*c*d^5*x^2+20592*C*a*b^2*c^2*d^4*x^2-6240*C*b^3*c^3*d^3*x^
2-7722*D*a^3*c*d^5*x^2+20592*D*a^2*b*c^2*d^4*x^2-18720*D*a*b^2*c^3*d^3*x^2+5760*D*b^3*c^4*d^2*x^2+45045*A*a^2*
b*d^6*x-36036*A*a*b^2*c*d^5*x+10296*A*b^3*c^2*d^4*x+15015*B*a^3*d^6*x-36036*B*a^2*b*c*d^5*x+30888*B*a*b^2*c^2*
d^4*x-9152*B*b^3*c^3*d^3*x-12012*C*a^3*c*d^5*x+30888*C*a^2*b*c^2*d^4*x-27456*C*a*b^2*c^3*d^3*x+8320*C*b^3*c^4*
d^2*x+10296*D*a^3*c^2*d^4*x-27456*D*a^2*b*c^3*d^3*x+24960*D*a*b^2*c^4*d^2*x-7680*D*b^3*c^5*d*x+45045*A*a^3*d^6
-90090*A*a^2*b*c*d^5+72072*A*a*b^2*c^2*d^4-20592*A*b^3*c^3*d^3-30030*B*a^3*c*d^5+72072*B*a^2*b*c^2*d^4-61776*B
*a*b^2*c^3*d^3+18304*B*b^3*c^4*d^2+24024*C*a^3*c^2*d^4-61776*C*a^2*b*c^3*d^3+54912*C*a*b^2*c^4*d^2-16640*C*b^3
*c^5*d-20592*D*a^3*c^3*d^3+54912*D*a^2*b*c^4*d^2-49920*D*a*b^2*c^5*d+15360*D*b^3*c^6)/d^7

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maxima [A]  time = 0.48, size = 621, normalized size = 1.42 \[ \frac {2 \, {\left (3465 \, {\left (d x + c\right )}^{\frac {13}{2}} D b^{3} - 4095 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 5005 \, {\left (15 \, D b^{3} c^{2} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 6435 \, {\left (20 \, D b^{3} c^{3} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 9009 \, {\left (15 \, D b^{3} c^{4} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 15015 \, {\left (6 \, D b^{3} c^{5} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 45045 \, {\left (D b^{3} c^{6} + A a^{3} d^{6} - {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )} \sqrt {d x + c}\right )}}{45045 \, d^{7}} \]

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)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(d*x + c)^(13/2)*D*b^3 - 4095*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d*x + c)^(11/2) + 5005*(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)^(9/2) - 6435*(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)^(7/2) + 9009*(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)^(5/2) - 15015*(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)^(3/2) + 45045*(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^7

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mupad [B]  time = 3.90, size = 713, normalized size = 1.64 \[ \frac {5544\,b^3\,c^6\,\sqrt {c+d\,x}\,D-504\,b^3\,c\,{\left (c+d\,x\right )}^{11/2}\,D-9240\,b^3\,c^5\,{\left (c+d\,x\right )}^{3/2}\,D+11088\,b^3\,c^4\,{\left (c+d\,x\right )}^{5/2}\,D-7920\,b^3\,c^3\,{\left (c+d\,x\right )}^{7/2}\,D+3080\,b^3\,c^2\,{\left (c+d\,x\right )}^{9/2}\,D+462\,b^3\,d^6\,x^6\,\sqrt {c+d\,x}\,D}{3003\,d^7}+\frac {2\,C\,{\left (c+d\,x\right )}^{5/2}\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{5\,d^6}+\frac {2\,A\,b^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,B\,b^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,C\,b^3\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {2\,A\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,A\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{d^4}+\frac {6\,A\,b^2\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {2\,B\,b^2\,\left (3\,a\,d-4\,b\,c\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}-\frac {2\,B\,c\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^5}+\frac {2\,C\,b^2\,\left (3\,a\,d-5\,b\,c\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {6\,B\,b\,{\left (c+d\,x\right )}^{5/2}\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{5\,d^5}+\frac {2\,C\,b\,{\left (c+d\,x\right )}^{7/2}\,\left (3\,a^2\,d^2-12\,a\,b\,c\,d+10\,b^2\,c^2\right )}{7\,d^6}+\frac {2\,B\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-4\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {2\,C\,c^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^6}-\frac {2\,a^3\,\sqrt {c+d\,x}\,D\,\left (6\,c\,{\left (c+d\,x\right )}^2-20\,c^2\,\left (c+d\,x\right )+30\,c^3-5\,d^3\,x^3\right )}{35\,d^4}-\frac {2\,a\,b^2\,\sqrt {c+d\,x}\,D\,\left (70\,c\,{\left (c+d\,x\right )}^4-840\,c^4\,\left (c+d\,x\right )-360\,c^2\,{\left (c+d\,x\right )}^3+756\,c^3\,{\left (c+d\,x\right )}^2+630\,c^5-63\,d^5\,x^5\right )}{231\,d^6}+\frac {2\,a^2\,b\,\sqrt {c+d\,x}\,D\,\left (168\,c^2\,{\left (c+d\,x\right )}^2-280\,c^3\,\left (c+d\,x\right )-40\,c\,{\left (c+d\,x\right )}^3+280\,c^4+35\,d^4\,x^4\right )}{105\,d^5}-\frac {2\,C\,c\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \]

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)^(1/2),x)

[Out]

(5544*b^3*c^6*(c + d*x)^(1/2)*D - 504*b^3*c*(c + d*x)^(11/2)*D - 9240*b^3*c^5*(c + d*x)^(3/2)*D + 11088*b^3*c^
4*(c + d*x)^(5/2)*D - 7920*b^3*c^3*(c + d*x)^(7/2)*D + 3080*b^3*c^2*(c + d*x)^(9/2)*D + 462*b^3*d^6*x^6*(c + d
*x)^(1/2)*D)/(3003*d^7) + (2*C*(c + d*x)^(5/2)*(a^3*d^3 - 10*b^3*c^3 + 18*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(5*d^6
) + (2*A*b^3*(c + d*x)^(7/2))/(7*d^4) + (2*B*b^3*(c + d*x)^(9/2))/(9*d^5) + (2*C*b^3*(c + d*x)^(11/2))/(11*d^6
) + (2*A*(a*d - b*c)^3*(c + d*x)^(1/2))/d^4 + (2*A*b*(a*d - b*c)^2*(c + d*x)^(3/2))/d^4 + (6*A*b^2*(a*d - b*c)
*(c + d*x)^(5/2))/(5*d^4) + (2*B*b^2*(3*a*d - 4*b*c)*(c + d*x)^(7/2))/(7*d^5) - (2*B*c*(a*d - b*c)^3*(c + d*x)
^(1/2))/d^5 + (2*C*b^2*(3*a*d - 5*b*c)*(c + d*x)^(9/2))/(9*d^6) + (6*B*b*(c + d*x)^(5/2)*(a^2*d^2 + 2*b^2*c^2
- 3*a*b*c*d))/(5*d^5) + (2*C*b*(c + d*x)^(7/2)*(3*a^2*d^2 + 10*b^2*c^2 - 12*a*b*c*d))/(7*d^6) + (2*B*(a*d - b*
c)^2*(a*d - 4*b*c)*(c + d*x)^(3/2))/(3*d^5) + (2*C*c^2*(a*d - b*c)^3*(c + d*x)^(1/2))/d^6 - (2*a^3*(c + d*x)^(
1/2)*D*(6*c*(c + d*x)^2 - 20*c^2*(c + d*x) + 30*c^3 - 5*d^3*x^3))/(35*d^4) - (2*a*b^2*(c + d*x)^(1/2)*D*(70*c*
(c + d*x)^4 - 840*c^4*(c + d*x) - 360*c^2*(c + d*x)^3 + 756*c^3*(c + d*x)^2 + 630*c^5 - 63*d^5*x^5))/(231*d^6)
+ (2*a^2*b*(c + d*x)^(1/2)*D*(168*c^2*(c + d*x)^2 - 280*c^3*(c + d*x) - 40*c*(c + d*x)^3 + 280*c^4 + 35*d^4*x
^4))/(105*d^5) - (2*C*c*(a*d - b*c)^2*(2*a*d - 5*b*c)*(c + d*x)^(3/2))/(3*d^6)

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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)**(1/2),x)

[Out]

Timed out

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