3.14.76 \(\int (a+b x)^4 \sqrt {c+d x} \, dx\) [1376]

Optimal. Leaf size=129 \[ \frac {2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \]

[Out]

2/3*(-a*d+b*c)^4*(d*x+c)^(3/2)/d^5-8/5*b*(-a*d+b*c)^3*(d*x+c)^(5/2)/d^5+12/7*b^2*(-a*d+b*c)^2*(d*x+c)^(7/2)/d^
5-8/9*b^3*(-a*d+b*c)*(d*x+c)^(9/2)/d^5+2/11*b^4*(d*x+c)^(11/2)/d^5

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Rubi [A]
time = 0.04, antiderivative size = 129, 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 = {45} \begin {gather*} -\frac {8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac {12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac {8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac {2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 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*(c + d*x)^(3/2))/(3*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(5/2))/(5*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(7/2))/(7*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(9/2))/(9*d^5) + (2*b^4*(c + d*x)^(11/2))/(11*d^5)

Rule 45

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 (a+b x)^4 \sqrt {c+d x} \, dx &=\int \left (\frac {(-b c+a d)^4 \sqrt {c+d x}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{3/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{5/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{7/2}}{d^4}+\frac {b^4 (c+d x)^{9/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 154, normalized size = 1.19 \begin {gather*} \frac {2 (c+d x)^{3/2} \left (1155 a^4 d^4+924 a^3 b d^3 (-2 c+3 d x)+198 a^2 b^2 d^2 \left (8 c^2-12 c d x+15 d^2 x^2\right )+44 a b^3 d \left (-16 c^3+24 c^2 d x-30 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (128 c^4-192 c^3 d x+240 c^2 d^2 x^2-280 c d^3 x^3+315 d^4 x^4\right )\right )}{3465 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(3/2)*(1155*a^4*d^4 + 924*a^3*b*d^3*(-2*c + 3*d*x) + 198*a^2*b^2*d^2*(8*c^2 - 12*c*d*x + 15*d^2*x
^2) + 44*a*b^3*d*(-16*c^3 + 24*c^2*d*x - 30*c*d^2*x^2 + 35*d^3*x^3) + b^4*(128*c^4 - 192*c^3*d*x + 240*c^2*d^2
*x^2 - 280*c*d^3*x^3 + 315*d^4*x^4)))/(3465*d^5)

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Maple [A]
time = 0.16, size = 100, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\)	\(100\)
default	\(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\)	\(100\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (315 d^{4} x^{4} b^{4}+1540 a \,b^{3} d^{4} x^{3}-280 b^{4} c \,d^{3} x^{3}+2970 a^{2} b^{2} d^{4} x^{2}-1320 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+2772 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1056 a \,b^{3} c^{2} d^{2} x -192 b^{4} c^{3} d x +1155 a^{4} d^{4}-1848 a^{3} b c \,d^{3}+1584 a^{2} b^{2} c^{2} d^{2}-704 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{3465 d^{5}}\)	\(186\)
trager	\(\frac {2 \left (315 b^{4} d^{5} x^{5}+1540 a \,b^{3} d^{5} x^{4}+35 b^{4} c \,d^{4} x^{4}+2970 a^{2} b^{2} d^{5} x^{3}+220 a \,b^{3} c \,d^{4} x^{3}-40 b^{4} c^{2} d^{3} x^{3}+2772 a^{3} b \,d^{5} x^{2}+594 a^{2} b^{2} c \,d^{4} x^{2}-264 a \,b^{3} c^{2} d^{3} x^{2}+48 b^{4} c^{3} d^{2} x^{2}+1155 a^{4} d^{5} x +924 a^{3} b c \,d^{4} x -792 a^{2} b^{2} c^{2} d^{3} x +352 a \,b^{3} c^{3} d^{2} x -64 b^{4} c^{4} d x +1155 a^{4} c \,d^{4}-1848 a^{3} b \,c^{2} d^{3}+1584 a^{2} b^{2} c^{3} d^{2}-704 a \,b^{3} c^{4} d +128 b^{4} c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\)	\(257\)
risch	\(\frac {2 \left (315 b^{4} d^{5} x^{5}+1540 a \,b^{3} d^{5} x^{4}+35 b^{4} c \,d^{4} x^{4}+2970 a^{2} b^{2} d^{5} x^{3}+220 a \,b^{3} c \,d^{4} x^{3}-40 b^{4} c^{2} d^{3} x^{3}+2772 a^{3} b \,d^{5} x^{2}+594 a^{2} b^{2} c \,d^{4} x^{2}-264 a \,b^{3} c^{2} d^{3} x^{2}+48 b^{4} c^{3} d^{2} x^{2}+1155 a^{4} d^{5} x +924 a^{3} b c \,d^{4} x -792 a^{2} b^{2} c^{2} d^{3} x +352 a \,b^{3} c^{3} d^{2} x -64 b^{4} c^{4} d x +1155 a^{4} c \,d^{4}-1848 a^{3} b \,c^{2} d^{3}+1584 a^{2} b^{2} c^{3} d^{2}-704 a \,b^{3} c^{4} d +128 b^{4} c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^5*(1/11*b^4*(d*x+c)^(11/2)+4/9*(a*d-b*c)*b^3*(d*x+c)^(9/2)+6/7*(a*d-b*c)^2*b^2*(d*x+c)^(7/2)+4/5*(a*d-b*c)
^3*b*(d*x+c)^(5/2)+1/3*(a*d-b*c)^4*(d*x+c)^(3/2))

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Maxima [A]
time = 0.29, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{3465 \, 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="maxima")

[Out]

2/3465*(315*(d*x + c)^(11/2)*b^4 - 1540*(b^4*c - a*b^3*d)*(d*x + c)^(9/2) + 2970*(b^4*c^2 - 2*a*b^3*c*d + a^2*
b^2*d^2)*(d*x + c)^(7/2) - 2772*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c)^(5/2) + 1155
*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(d*x + c)^(3/2))/d^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (109) = 218\).
time = 0.43, size = 245, normalized size = 1.90 \begin {gather*} \frac {2 \, {\left (315 \, b^{4} d^{5} x^{5} + 128 \, b^{4} c^{5} - 704 \, a b^{3} c^{4} d + 1584 \, a^{2} b^{2} c^{3} d^{2} - 1848 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} + 35 \, {\left (b^{4} c d^{4} + 44 \, a b^{3} d^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} - 297 \, a^{2} b^{2} d^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} c^{3} d^{2} - 44 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} + 462 \, a^{3} b d^{5}\right )} x^{2} - {\left (64 \, b^{4} c^{4} d - 352 \, a b^{3} c^{3} d^{2} + 792 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} - 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, 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]

2/3465*(315*b^4*d^5*x^5 + 128*b^4*c^5 - 704*a*b^3*c^4*d + 1584*a^2*b^2*c^3*d^2 - 1848*a^3*b*c^2*d^3 + 1155*a^4
*c*d^4 + 35*(b^4*c*d^4 + 44*a*b^3*d^5)*x^4 - 10*(4*b^4*c^2*d^3 - 22*a*b^3*c*d^4 - 297*a^2*b^2*d^5)*x^3 + 6*(8*
b^4*c^3*d^2 - 44*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 + 462*a^3*b*d^5)*x^2 - (64*b^4*c^4*d - 352*a*b^3*c^3*d^2 + 7
92*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 - 1155*a^4*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]
time = 1.87, size = 223, normalized size = 1.73 \begin {gather*} \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (4 a b^{3} d - 4 b^{4} c\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 d^{4}}\right )}{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)

[Out]

2*(b**4*(c + d*x)**(11/2)/(11*d**4) + (c + d*x)**(9/2)*(4*a*b**3*d - 4*b**4*c)/(9*d**4) + (c + d*x)**(7/2)*(6*
a**2*b**2*d**2 - 12*a*b**3*c*d + 6*b**4*c**2)/(7*d**4) + (c + d*x)**(5/2)*(4*a**3*b*d**3 - 12*a**2*b**2*c*d**2
+ 12*a*b**3*c**2*d - 4*b**4*c**3)/(5*d**4) + (c + d*x)**(3/2)*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2
*d**2 - 4*a*b**3*c**3*d + b**4*c**4)/(3*d**4))/d

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).
time = 1.21, size = 470, normalized size = 3.64 \begin {gather*} \frac {2 \, {\left (3465 \, \sqrt {d x + c} a^{4} c + 1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} + \frac {4620 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b c}{d} + \frac {1386 \, {\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} c}{d^{2}} + \frac {924 \, {\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^{3} b}{d} + \frac {396 \, {\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} c}{d^{3}} + \frac {594 \, {\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^{2} b^{2}}{d^{2}} + \frac {11 \, {\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} c}{d^{4}} + \frac {44 \, {\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 )} a b^{3}}{d^{3}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{4}}{d^{4}}\right )}}{3465 \, 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]

2/3465*(3465*sqrt(d*x + c)*a^4*c + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4 + 4620*((d*x + c)^(3/2) - 3*
sqrt(d*x + c)*c)*a^3*b*c/d + 1386*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b^2*c/
d^2 + 924*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b/d + 396*(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*c/d^3 + 594*(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^2*b^2/d^2 + 11*(35*(d*x + c)^(9/2) - 1
80*(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*c/d^4 +
44*(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)*a*b^3/d^3 + 5*(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)*b^4/d^4)/d

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Mupad [B]
time = 0.22, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{5\,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)^(11/2))/(11*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(9/2))/(9*d^5) + (2*(a*d - b*c)^4*(c + d*
x)^(3/2))/(3*d^5) + (12*b^2*(a*d - b*c)^2*(c + d*x)^(7/2))/(7*d^5) + (8*b*(a*d - b*c)^3*(c + d*x)^(5/2))/(5*d^
5)

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