3.1.12 \(\int \frac {(a+b x) (A+B x+C x^2+D x^3)}{(c+d x)^{3/2}} \, dx\) [12]

Optimal. Leaf size=210 \[ \frac {2 (b c-a d) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^5 \sqrt {c+d x}}-\frac {2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) \sqrt {c+d x}}{d^5}+\frac {2 \left (a d (C d-3 c D)-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) (c+d x)^{3/2}}{3 d^5}+\frac {2 (b C d-4 b c D+a d D) (c+d x)^{5/2}}{5 d^5}+\frac {2 b D (c+d x)^{7/2}}{7 d^5} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1634} \begin {gather*} -\frac {2 \sqrt {c+d x} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5}+\frac {2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 \sqrt {c+d x}}+\frac {2 (c+d x)^{3/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac {2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac {2 b D (c+d x)^{7/2}}{7 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.20, size = 188, normalized size = 0.90 \begin {gather*} \frac {14 a d \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 )+b \left (-768 c^4 D+96 c^3 d (7 C-4 D x)+16 c^2 d^2 (-35 B+3 x (7 C+2 D x))+4 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+2 d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))\right )}{105 d^5 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*a*d*(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))) + b*(-768*c^4*D + 96*c^3*d*(7*C - 4*D*x) + 16*c^2*d^2*(-35*B + 3*x*(7*C + 2*D*x)) + 4*c*d^3*(105*A
- x*(70*B + 3*x*(7*C + 4*D*x))) + 2*d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))))/(105*d^5*Sqrt[c + d*x])

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Maple [A]
time = 0.10, size = 294, normalized size = 1.40

method	result	size
gosper	\(-\frac {2 \left (-15 D b \,x^{4} d^{4}-21 C b \,d^{4} x^{3}-21 D a \,d^{4} x^{3}+24 D b c \,d^{3} x^{3}-35 B b \,d^{4} x^{2}-35 C a \,d^{4} x^{2}+42 C b c \,d^{3} x^{2}+42 D a c \,d^{3} x^{2}-48 D b \,c^{2} d^{2} x^{2}-105 A b \,d^{4} x -105 B a \,d^{4} x +140 B b c \,d^{3} x +140 C a c \,d^{3} x -168 C b \,c^{2} d^{2} x -168 D a \,c^{2} d^{2} x +192 D b \,c^{3} d x +105 A a \,d^{4}-210 A b c \,d^{3}-210 B a c \,d^{3}+280 B b \,c^{2} d^{2}+280 C a \,c^{2} d^{2}-336 C b \,c^{3} d -336 D a \,c^{3} d +384 D b \,c^{4}\right )}{105 \sqrt {d x +c}\, d^{5}}\)	\(241\)
trager	\(-\frac {2 \left (-15 D b \,x^{4} d^{4}-21 C b \,d^{4} x^{3}-21 D a \,d^{4} x^{3}+24 D b c \,d^{3} x^{3}-35 B b \,d^{4} x^{2}-35 C a \,d^{4} x^{2}+42 C b c \,d^{3} x^{2}+42 D a c \,d^{3} x^{2}-48 D b \,c^{2} d^{2} x^{2}-105 A b \,d^{4} x -105 B a \,d^{4} x +140 B b c \,d^{3} x +140 C a c \,d^{3} x -168 C b \,c^{2} d^{2} x -168 D a \,c^{2} d^{2} x +192 D b \,c^{3} d x +105 A a \,d^{4}-210 A b c \,d^{3}-210 B a c \,d^{3}+280 B b \,c^{2} d^{2}+280 C a \,c^{2} d^{2}-336 C b \,c^{3} d -336 D a \,c^{3} d +384 D b \,c^{4}\right )}{105 \sqrt {d x +c}\, d^{5}}\)	\(241\)
derivativedivides	\(\frac {\frac {2 D b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C b d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a d \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 D b c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 C b c d \left (d x +c \right )^{\frac {3}{2}}-2 D a c d \left (d x +c \right )^{\frac {3}{2}}+4 D b \,c^{2} \left (d x +c \right )^{\frac {3}{2}}+2 A \,d^{3} b \sqrt {d x +c}+2 B a \,d^{3} \sqrt {d x +c}-4 B b c \,d^{2} \sqrt {d x +c}-4 C a c \,d^{2} \sqrt {d x +c}+6 C \,c^{2} d b \sqrt {d x +c}+6 D a \,c^{2} d \sqrt {d x +c}-8 D c^{3} b \sqrt {d x +c}-\frac {2 \left (A a \,d^{4}-A b c \,d^{3}-B a c \,d^{3}+B b \,c^{2} d^{2}+C a \,c^{2} d^{2}-C b \,c^{3} d -D a \,c^{3} d +D b \,c^{4}\right )}{\sqrt {d x +c}}}{d^{5}}\)	\(294\)
default	\(\frac {\frac {2 D b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 C b d \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a d \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 D b c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B b \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 C b c d \left (d x +c \right )^{\frac {3}{2}}-2 D a c d \left (d x +c \right )^{\frac {3}{2}}+4 D b \,c^{2} \left (d x +c \right )^{\frac {3}{2}}+2 A \,d^{3} b \sqrt {d x +c}+2 B a \,d^{3} \sqrt {d x +c}-4 B b c \,d^{2} \sqrt {d x +c}-4 C a c \,d^{2} \sqrt {d x +c}+6 C \,c^{2} d b \sqrt {d x +c}+6 D a \,c^{2} d \sqrt {d x +c}-8 D c^{3} b \sqrt {d x +c}-\frac {2 \left (A a \,d^{4}-A b c \,d^{3}-B a c \,d^{3}+B b \,c^{2} d^{2}+C a \,c^{2} d^{2}-C b \,c^{3} d -D a \,c^{3} d +D b \,c^{4}\right )}{\sqrt {d x +c}}}{d^{5}}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^5*(1/7*D*b*(d*x+c)^(7/2)+1/5*C*b*d*(d*x+c)^(5/2)+1/5*D*a*d*(d*x+c)^(5/2)-4/5*D*b*c*(d*x+c)^(5/2)+1/3*B*b*d
^2*(d*x+c)^(3/2)+1/3*C*a*d^2*(d*x+c)^(3/2)-C*b*c*d*(d*x+c)^(3/2)-D*a*c*d*(d*x+c)^(3/2)+2*D*b*c^2*(d*x+c)^(3/2)
+A*d^3*b*(d*x+c)^(1/2)+B*a*d^3*(d*x+c)^(1/2)-2*B*b*c*d^2*(d*x+c)^(1/2)-2*C*a*c*d^2*(d*x+c)^(1/2)+3*C*c^2*d*b*(
d*x+c)^(1/2)+3*D*a*c^2*d*(d*x+c)^(1/2)-4*D*c^3*b*(d*x+c)^(1/2)-(A*a*d^4-A*b*c*d^3-B*a*c*d^3+B*b*c^2*d^2+C*a*c^
2*d^2-C*b*c^3*d-D*a*c^3*d+D*b*c^4)/(d*x+c)^(1/2))

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Maxima [A]
time = 0.33, size = 206, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b - 21 \, {\left (4 \, D b c - {\left (D a + C b\right )} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, D b c^{2} - 3 \, {\left (D a + C b\right )} c d + {\left (C a + B b\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 105 \, {\left (4 \, D b c^{3} - 3 \, {\left (D a + C b\right )} c^{2} d + 2 \, {\left (C a + B b\right )} c d^{2} - {\left (B a + A b\right )} d^{3}\right )} \sqrt {d x + c}}{d^{4}} - \frac {105 \, {\left (D b c^{4} + A a d^{4} - {\left (D a + C b\right )} c^{3} d + {\left (C a + B b\right )} c^{2} d^{2} - {\left (B a + A b\right )} c d^{3}\right )}}{\sqrt {d x + c} d^{4}}\right )}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((15*(d*x + c)^(7/2)*D*b - 21*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(5/2) + 35*(6*D*b*c^2 - 3*(D*a + C*b)*
c*d + (C*a + B*b)*d^2)*(d*x + c)^(3/2) - 105*(4*D*b*c^3 - 3*(D*a + C*b)*c^2*d + 2*(C*a + B*b)*c*d^2 - (B*a + A
*b)*d^3)*sqrt(d*x + c))/d^4 - 105*(D*b*c^4 + A*a*d^4 - (D*a + C*b)*c^3*d + (C*a + B*b)*c^2*d^2 - (B*a + A*b)*c
*d^3)/(sqrt(d*x + c)*d^4))/d

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Fricas [A]
time = 0.69, size = 213, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (15 \, D b d^{4} x^{4} - 384 \, D b c^{4} - 105 \, A a d^{4} - 280 \, {\left (C a + B b\right )} c^{2} d^{2} + 210 \, {\left (B a + A b\right )} c d^{3} - 3 \, {\left (8 \, D b c d^{3} - 7 \, {\left (D a + C b\right )} d^{4}\right )} x^{3} + {\left (48 \, D b c^{2} d^{2} + 35 \, {\left (C a + B b\right )} d^{4} - 42 \, {\left (D a c + C b c\right )} d^{3}\right )} x^{2} + 336 \, {\left (D a c^{3} + C b c^{3}\right )} d - {\left (192 \, D b c^{3} d + 140 \, {\left (C a + B b\right )} c d^{3} - 105 \, {\left (B a + A b\right )} d^{4} - 168 \, {\left (D a c^{2} + C b c^{2}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (d^{6} x + c d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*D*b*d^4*x^4 - 384*D*b*c^4 - 105*A*a*d^4 - 280*(C*a + B*b)*c^2*d^2 + 210*(B*a + A*b)*c*d^3 - 3*(8*D*b
*c*d^3 - 7*(D*a + C*b)*d^4)*x^3 + (48*D*b*c^2*d^2 + 35*(C*a + B*b)*d^4 - 42*(D*a*c + C*b*c)*d^3)*x^2 + 336*(D*
a*c^3 + C*b*c^3)*d - (192*D*b*c^3*d + 140*(C*a + B*b)*c*d^3 - 105*(B*a + A*b)*d^4 - 168*(D*a*c^2 + C*b*c^2)*d^
2)*x)*sqrt(d*x + c)/(d^6*x + c*d^5)

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Sympy [A]
time = 21.15, size = 230, normalized size = 1.10 \begin {gather*} \frac {2 D b \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 C b d + 2 D a d - 8 D b c\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (2 B b d^{2} + 2 C a d^{2} - 6 C b c d - 6 D a c d + 12 D b c^{2}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (2 A b d^{3} + 2 B a d^{3} - 4 B b c d^{2} - 4 C a c d^{2} + 6 C b c^{2} d + 6 D a c^{2} d - 8 D b c^{3}\right )}{d^{5}} + \frac {2 \left (a d - b c\right ) \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{5} \sqrt {c + d x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*D*b*(c + d*x)**(7/2)/(7*d**5) + (c + d*x)**(5/2)*(2*C*b*d + 2*D*a*d - 8*D*b*c)/(5*d**5) + (c + d*x)**(3/2)*(
2*B*b*d**2 + 2*C*a*d**2 - 6*C*b*c*d - 6*D*a*c*d + 12*D*b*c**2)/(3*d**5) + sqrt(c + d*x)*(2*A*b*d**3 + 2*B*a*d*
*3 - 4*B*b*c*d**2 - 4*C*a*c*d**2 + 6*C*b*c**2*d + 6*D*a*c**2*d - 8*D*b*c**3)/d**5 + 2*(a*d - b*c)*(-A*d**3 + B
*c*d**2 - C*c**2*d + D*c**3)/(d**5*sqrt(c + d*x))

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Giac [A]
time = 0.57, size = 323, normalized size = 1.54 \begin {gather*} -\frac {2 \, {\left (D b c^{4} - D a c^{3} d - C b c^{3} d + C a c^{2} d^{2} + B b c^{2} d^{2} - B a c d^{3} - A b c d^{3} + A a d^{4}\right )}}{\sqrt {d x + c} d^{5}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b d^{30} - 84 \, {\left (d x + c\right )}^{\frac {5}{2}} D b c d^{30} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} D b c^{2} d^{30} - 420 \, \sqrt {d x + c} D b c^{3} d^{30} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} D a d^{31} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} C b d^{31} - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} D a c d^{31} - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} C b c d^{31} + 315 \, \sqrt {d x + c} D a c^{2} d^{31} + 315 \, \sqrt {d x + c} C b c^{2} d^{31} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} C a d^{32} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} B b d^{32} - 210 \, \sqrt {d x + c} C a c d^{32} - 210 \, \sqrt {d x + c} B b c d^{32} + 105 \, \sqrt {d x + c} B a d^{33} + 105 \, \sqrt {d x + c} A b d^{33}\right )}}{105 \, d^{35}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(D*b*c^4 - D*a*c^3*d - C*b*c^3*d + C*a*c^2*d^2 + B*b*c^2*d^2 - B*a*c*d^3 - A*b*c*d^3 + A*a*d^4)/(sqrt(d*x +
c)*d^5) + 2/105*(15*(d*x + c)^(7/2)*D*b*d^30 - 84*(d*x + c)^(5/2)*D*b*c*d^30 + 210*(d*x + c)^(3/2)*D*b*c^2*d^
30 - 420*sqrt(d*x + c)*D*b*c^3*d^30 + 21*(d*x + c)^(5/2)*D*a*d^31 + 21*(d*x + c)^(5/2)*C*b*d^31 - 105*(d*x + c
)^(3/2)*D*a*c*d^31 - 105*(d*x + c)^(3/2)*C*b*c*d^31 + 315*sqrt(d*x + c)*D*a*c^2*d^31 + 315*sqrt(d*x + c)*C*b*c
^2*d^31 + 35*(d*x + c)^(3/2)*C*a*d^32 + 35*(d*x + c)^(3/2)*B*b*d^32 - 210*sqrt(d*x + c)*C*a*c*d^32 - 210*sqrt(
d*x + c)*B*b*c*d^32 + 105*sqrt(d*x + c)*B*a*d^33 + 105*sqrt(d*x + c)*A*b*d^33)/d^35

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

[Out]

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

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