3.14.8 \(\int \frac {(a+b x)^4}{\sqrt {c+d x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac {8 b^3 (c+d x)^{7/2} (b c-a d)}{7 d^5}+\frac {12 b^2 (c+d x)^{5/2} (b c-a d)^2}{5 d^5}-\frac {8 b (c+d x)^{3/2} (b c-a d)^3}{3 d^5}+\frac {2 \sqrt {c+d x} (b c-a d)^4}{d^5}+\frac {2 b^4 (c+d x)^{9/2}}{9 d^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} -\frac {8 b^3 (c+d x)^{7/2} (b c-a d)}{7 d^5}+\frac {12 b^2 (c+d x)^{5/2} (b c-a d)^2}{5 d^5}-\frac {8 b (c+d x)^{3/2} (b c-a d)^3}{3 d^5}+\frac {2 \sqrt {c+d x} (b c-a d)^4}{d^5}+\frac {2 b^4 (c+d x)^{9/2}}{9 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/Sqrt[c + d*x],x]

[Out]

(2*(b*c - a*d)^4*Sqrt[c + d*x])/d^5 - (8*b*(b*c - a*d)^3*(c + d*x)^(3/2))/(3*d^5) + (12*b^2*(b*c - a*d)^2*(c +
d*x)^(5/2))/(5*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^5) + (2*b^4*(c + d*x)^(9/2))/(9*d^5)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x)^4}{\sqrt {c+d x}} \, dx &=\int \left (\frac {(-b c+a d)^4}{d^4 \sqrt {c+d x}}-\frac {4 b (b c-a d)^3 \sqrt {c+d x}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{5/2}}{d^4}+\frac {b^4 (c+d x)^{7/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 \sqrt {c+d x}}{d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{3/2}}{3 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{5/2}}{5 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{7/2}}{7 d^5}+\frac {2 b^4 (c+d x)^{9/2}}{9 d^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 101, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {c+d x} \left (-180 b^3 (c+d x)^3 (b c-a d)+378 b^2 (c+d x)^2 (b c-a d)^2-420 b (c+d x) (b c-a d)^3+315 (b c-a d)^4+35 b^4 (c+d x)^4\right )}{315 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(315*(b*c - a*d)^4 - 420*b*(b*c - a*d)^3*(c + d*x) + 378*b^2*(b*c - a*d)^2*(c + d*x)^2 - 180*
b^3*(b*c - a*d)*(c + d*x)^3 + 35*b^4*(c + d*x)^4))/(315*d^5)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 213, normalized size = 1.68 \begin {gather*} \frac {2 \sqrt {c+d x} \left (315 a^4 d^4+420 a^3 b d^3 (c+d x)-1260 a^3 b c d^3+1890 a^2 b^2 c^2 d^2+378 a^2 b^2 d^2 (c+d x)^2-1260 a^2 b^2 c d^2 (c+d x)-1260 a b^3 c^3 d+1260 a b^3 c^2 d (c+d x)+180 a b^3 d (c+d x)^3-756 a b^3 c d (c+d x)^2+315 b^4 c^4-420 b^4 c^3 (c+d x)+378 b^4 c^2 (c+d x)^2+35 b^4 (c+d x)^4-180 b^4 c (c+d x)^3\right )}{315 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^4/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(315*b^4*c^4 - 1260*a*b^3*c^3*d + 1890*a^2*b^2*c^2*d^2 - 1260*a^3*b*c*d^3 + 315*a^4*d^4 - 420
*b^4*c^3*(c + d*x) + 1260*a*b^3*c^2*d*(c + d*x) - 1260*a^2*b^2*c*d^2*(c + d*x) + 420*a^3*b*d^3*(c + d*x) + 378
*b^4*c^2*(c + d*x)^2 - 756*a*b^3*c*d*(c + d*x)^2 + 378*a^2*b^2*d^2*(c + d*x)^2 - 180*b^4*c*(c + d*x)^3 + 180*a
*b^3*d*(c + d*x)^3 + 35*b^4*(c + d*x)^4))/(315*d^5)

________________________________________________________________________________________

fricas [A]  time = 1.18, size = 182, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c^{4} - 576 \, a b^{3} c^{3} d + 1008 \, a^{2} b^{2} c^{2} d^{2} - 840 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 20 \, {\left (2 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (8 \, b^{4} c^{2} d^{2} - 36 \, a b^{3} c d^{3} + 63 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} c^{3} d - 72 \, a b^{3} c^{2} d^{2} + 126 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*d^4*x^4 + 128*b^4*c^4 - 576*a*b^3*c^3*d + 1008*a^2*b^2*c^2*d^2 - 840*a^3*b*c*d^3 + 315*a^4*d^4 -
20*(2*b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 6*(8*b^4*c^2*d^2 - 36*a*b^3*c*d^3 + 63*a^2*b^2*d^4)*x^2 - 4*(16*b^4*c^3*
d - 72*a*b^3*c^2*d^2 + 126*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(d*x + c)/d^5

________________________________________________________________________________________

giac [A]  time = 0.96, size = 204, normalized size = 1.61 \begin {gather*} \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{4} + \frac {420 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b}{d} + \frac {126 \, {\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 )} a^{2} b^{2}}{d^{2}} + \frac {36 \, {\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 )} a b^{3}}{d^{3}} + \frac {{\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{4}}{d^{4}}\right )}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(d*x + c)*a^4 + 420*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*b/d + 126*(3*(d*x + c)^(5/2) - 10
*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2/d^2 + 36*(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 + (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^4/d^4)/d

________________________________________________________________________________________

maple [A]  time = 0.01, size = 186, normalized size = 1.46 \begin {gather*} \frac {2 \sqrt {d x +c}\, \left (35 b^{4} x^{4} d^{4}+180 a \,b^{3} d^{4} x^{3}-40 b^{4} c \,d^{3} x^{3}+378 a^{2} b^{2} d^{4} x^{2}-216 a \,b^{3} c \,d^{3} x^{2}+48 b^{4} c^{2} d^{2} x^{2}+420 a^{3} b \,d^{4} x -504 a^{2} b^{2} c \,d^{3} x +288 a \,b^{3} c^{2} d^{2} x -64 b^{4} c^{3} d x +315 a^{4} d^{4}-840 a^{3} b c \,d^{3}+1008 a^{2} b^{2} c^{2} d^{2}-576 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{315 d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c)^(1/2),x)

[Out]

2/315*(d*x+c)^(1/2)*(35*b^4*d^4*x^4+180*a*b^3*d^4*x^3-40*b^4*c*d^3*x^3+378*a^2*b^2*d^4*x^2-216*a*b^3*c*d^3*x^2
+48*b^4*c^2*d^2*x^2+420*a^3*b*d^4*x-504*a^2*b^2*c*d^3*x+288*a*b^3*c^2*d^2*x-64*b^4*c^3*d*x+315*a^4*d^4-840*a^3
*b*c*d^3+1008*a^2*b^2*c^2*d^2-576*a*b^3*c^3*d+128*b^4*c^4)/d^5

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 204, normalized size = 1.61 \begin {gather*} \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{4} + \frac {420 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b}{d} + \frac {126 \, {\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 )} a^{2} b^{2}}{d^{2}} + \frac {36 \, {\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 )} a b^{3}}{d^{3}} + \frac {{\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{4}}{d^{4}}\right )}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(d*x + c)*a^4 + 420*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*b/d + 126*(3*(d*x + c)^(5/2) - 10
*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2/d^2 + 36*(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 + (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^4/d^4)/d

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 112, normalized size = 0.88 \begin {gather*} \frac {2\,b^4\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,\sqrt {c+d\,x}}{d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(c + d*x)^(1/2),x)

[Out]

(2*b^4*(c + d*x)^(9/2))/(9*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(7/2))/(7*d^5) + (2*(a*d - b*c)^4*(c + d*x)
^(1/2))/d^5 + (12*b^2*(a*d - b*c)^2*(c + d*x)^(5/2))/(5*d^5) + (8*b*(a*d - b*c)^3*(c + d*x)^(3/2))/(3*d^5)

________________________________________________________________________________________

sympy [A]  time = 56.90, size = 532, normalized size = 4.19 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{4} c}{\sqrt {c + d x}} - 2 a^{4} \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {8 a^{3} b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {8 a^{3} 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 {12 a^{2} b^{2} 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 {12 a^{2} b^{2} \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 {8 a b^{3} 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 {8 a b^{3} \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}} - \frac {2 b^{4} c \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^{4}} - \frac {2 b^{4} \left (- \frac {c^{5}}{\sqrt {c + d x}} - 5 c^{4} \sqrt {c + d x} + \frac {10 c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} - 2 c^{2} \left (c + d x\right )^{\frac {5}{2}} + \frac {5 c \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}}}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(d*x+c)**(1/2),x)

[Out]

Piecewise(((-2*a**4*c/sqrt(c + d*x) - 2*a**4*(-c/sqrt(c + d*x) - sqrt(c + d*x)) - 8*a**3*b*c*(-c/sqrt(c + d*x)
- sqrt(c + d*x))/d - 8*a**3*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d - 12*a**2*b**2*
c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 - 12*a**2*b**2*(-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 - 8*a*b**3*c*(-c**3/sqrt(c + d*x) - 3*c**2*s
qrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**3 - 8*a*b**3*(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 - 2*b**4*c*(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**4 - 2*b
**4*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(
c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4)/d, Ne(d, 0)), (Piecewise((a**4*x, Eq(b, 0)), ((a + b*x)**5/(5*b)
, True))/sqrt(c), True))

________________________________________________________________________________________