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

Optimal. Leaf size=115 \[ \frac {2 \sqrt {c+d x} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4}-\frac {2 (c+d x)^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^4}+\frac {2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac {2 D (c+d x)^{7/2}}{7 d^4} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1850} \[ \frac {2 \sqrt {c+d x} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4}-\frac {2 (c+d x)^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^4}+\frac {2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac {2 D (c+d x)^{7/2}}{7 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 82, normalized size = 0.71 \[ \frac {2 \sqrt {c+d x} \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right )}{105 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105*A + x*(35*B
+ 3*x*(7*C + 5*D*x)))))/(105*d^4)

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fricas [A]  time = 0.54, size = 90, normalized size = 0.78 \[ \frac {2 \, {\left (15 \, D d^{3} x^{3} - 48 \, D c^{3} + 56 \, C c^{2} d - 70 \, B c d^{2} + 105 \, A d^{3} - 3 \, {\left (6 \, D c d^{2} - 7 \, C d^{3}\right )} x^{2} + {\left (24 \, D c^{2} d - 28 \, C c d^{2} + 35 \, B d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*D*d^3*x^3 - 48*D*c^3 + 56*C*c^2*d - 70*B*c*d^2 + 105*A*d^3 - 3*(6*D*c*d^2 - 7*C*d^3)*x^2 + (24*D*c^2
*d - 28*C*c*d^2 + 35*B*d^3)*x)*sqrt(d*x + c)/d^4

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giac [A]  time = 1.21, size = 128, normalized size = 1.11 \[ \frac {2 \, {\left (105 \, \sqrt {d x + c} A + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} B}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} C}{d^{2}} + \frac {3 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} D}{d^{3}}\right )}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/105*(105*sqrt(d*x + c)*A + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C/d^2 + 3*(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/d^3)/d

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maple [A]  time = 0.01, size = 91, normalized size = 0.79 \[ \frac {2 \sqrt {d x +c}\, \left (15 D x^{3} d^{3}+21 C \,d^{3} x^{2}-18 D c \,d^{2} x^{2}+35 B \,d^{3} x -28 C c \,d^{2} x +24 D c^{2} d x +105 A \,d^{3}-70 B c \,d^{2}+56 C \,c^{2} d -48 D c^{3}\right )}{105 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)

[Out]

2/105*(d*x+c)^(1/2)*(15*D*d^3*x^3+21*C*d^3*x^2-18*D*c*d^2*x^2+35*B*d^3*x-28*C*c*d^2*x+24*D*c^2*d*x+105*A*d^3-7
0*B*c*d^2+56*C*c^2*d-48*D*c^3)/d^4

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maxima [A]  time = 0.44, size = 128, normalized size = 1.11 \[ \frac {2 \, {\left (105 \, \sqrt {d x + c} A + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} B}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} C}{d^{2}} + \frac {3 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} D}{d^{3}}\right )}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/105*(105*sqrt(d*x + c)*A + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C/d^2 + 3*(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/d^3)/d

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mupad [B]  time = 2.85, size = 128, normalized size = 1.11 \[ \frac {6\,C\,{\left (c+d\,x\right )}^{5/2}-20\,C\,c\,{\left (c+d\,x\right )}^{3/2}+30\,C\,c^2\,\sqrt {c+d\,x}}{15\,d^3}+\frac {2\,B\,{\left (c+d\,x\right )}^{3/2}-6\,B\,c\,\sqrt {c+d\,x}}{3\,d^2}+\frac {2\,A\,\sqrt {c+d\,x}}{d}-\frac {2\,\sqrt {c+d\,x}\,D\,\left (6\,c\,{\left (c+d\,x\right )}^2-20\,c^2\,\left (c+d\,x\right )+30\,c^3-5\,d^3\,x^3\right )}{35\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2 + x^3*D)/(c + d*x)^(1/2),x)

[Out]

(6*C*(c + d*x)^(5/2) - 20*C*c*(c + d*x)^(3/2) + 30*C*c^2*(c + d*x)^(1/2))/(15*d^3) + (2*B*(c + d*x)^(3/2) - 6*
B*c*(c + d*x)^(1/2))/(3*d^2) + (2*A*(c + d*x)^(1/2))/d - (2*(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)

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sympy [A]  time = 18.09, size = 354, normalized size = 3.08 \[ \begin {cases} \frac {- \frac {2 A c}{\sqrt {c + d x}} - 2 A \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {2 B c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {2 B \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d} - \frac {2 C c \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} - \frac {2 C \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} - \frac {2 D c \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{3}} - \frac {2 D \left (\frac {c^{4}}{\sqrt {c + d x}} + 4 c^{3} \sqrt {c + d x} - 2 c^{2} \left (c + d x\right )^{\frac {3}{2}} + \frac {4 c \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}}}{d} & \text {for}\: d \neq 0 \\\frac {A x + \frac {B x^{2}}{2} + \frac {C x^{3}}{3} + \frac {D x^{4}}{4}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)

[Out]

Piecewise(((-2*A*c/sqrt(c + d*x) - 2*A*(-c/sqrt(c + d*x) - sqrt(c + d*x)) - 2*B*c*(-c/sqrt(c + d*x) - sqrt(c +
d*x))/d - 2*B*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d - 2*C*c*(c**2/sqrt(c + d*x) + 2
*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 - 2*C*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(
3/2) - (c + d*x)**(5/2)/5)/d**2 - 2*D*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c
+ d*x)**(5/2)/5)/d**3 - 2*D*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*
x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**3)/d, Ne(d, 0)), ((A*x + B*x**2/2 + C*x**3/3 + D*x**4/4)/sqrt(c), True))

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