3.15.13 \(\int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx\) [1413]

Optimal. Leaf size=154 \[ -\frac {2 (b c-a d)^5 \sqrt {c+d x}}{d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{3/2}}{3 d^6}-\frac {4 b^2 (b c-a d)^3 (c+d x)^{5/2}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{7/2}}{7 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{9/2}}{9 d^6}+\frac {2 b^5 (c+d x)^{11/2}}{11 d^6} \]

[Out]

10/3*b*(-a*d+b*c)^4*(d*x+c)^(3/2)/d^6-4*b^2*(-a*d+b*c)^3*(d*x+c)^(5/2)/d^6+20/7*b^3*(-a*d+b*c)^2*(d*x+c)^(7/2)
/d^6-10/9*b^4*(-a*d+b*c)*(d*x+c)^(9/2)/d^6+2/11*b^5*(d*x+c)^(11/2)/d^6-2*(-a*d+b*c)^5*(d*x+c)^(1/2)/d^6

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Rubi [A]
time = 0.03, antiderivative size = 154, 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 {10 b^4 (c+d x)^{9/2} (b c-a d)}{9 d^6}+\frac {20 b^3 (c+d x)^{7/2} (b c-a d)^2}{7 d^6}-\frac {4 b^2 (c+d x)^{5/2} (b c-a d)^3}{d^6}+\frac {10 b (c+d x)^{3/2} (b c-a d)^4}{3 d^6}-\frac {2 \sqrt {c+d x} (b c-a d)^5}{d^6}+\frac {2 b^5 (c+d x)^{11/2}}{11 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/Sqrt[c + d*x],x]

[Out]

(-2*(b*c - a*d)^5*Sqrt[c + d*x])/d^6 + (10*b*(b*c - a*d)^4*(c + d*x)^(3/2))/(3*d^6) - (4*b^2*(b*c - a*d)^3*(c
+ d*x)^(5/2))/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x)^(7/2))/(7*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^(9/2))/(9*d
^6) + (2*b^5*(c + d*x)^(11/2))/(11*d^6)

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 \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx &=\int \left (\frac {(-b c+a d)^5}{d^5 \sqrt {c+d x}}+\frac {5 b (b c-a d)^4 \sqrt {c+d x}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{3/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{5/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{7/2}}{d^5}+\frac {b^5 (c+d x)^{9/2}}{d^5}\right ) \, dx\\ &=-\frac {2 (b c-a d)^5 \sqrt {c+d x}}{d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{3/2}}{3 d^6}-\frac {4 b^2 (b c-a d)^3 (c+d x)^{5/2}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{7/2}}{7 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{9/2}}{9 d^6}+\frac {2 b^5 (c+d x)^{11/2}}{11 d^6}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 216, normalized size = 1.40 \begin {gather*} \frac {2 \sqrt {c+d x} \left (693 a^5 d^5+1155 a^4 b d^4 (-2 c+d x)+462 a^3 b^2 d^3 \left (8 c^2-4 c d x+3 d^2 x^2\right )+198 a^2 b^3 d^2 \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )+11 a b^4 d \left (128 c^4-64 c^3 d x+48 c^2 d^2 x^2-40 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (-256 c^5+128 c^4 d x-96 c^3 d^2 x^2+80 c^2 d^3 x^3-70 c d^4 x^4+63 d^5 x^5\right )\right )}{693 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(693*a^5*d^5 + 1155*a^4*b*d^4*(-2*c + d*x) + 462*a^3*b^2*d^3*(8*c^2 - 4*c*d*x + 3*d^2*x^2) +
198*a^2*b^3*d^2*(-16*c^3 + 8*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3) + 11*a*b^4*d*(128*c^4 - 64*c^3*d*x + 48*c^2*d^
2*x^2 - 40*c*d^3*x^3 + 35*d^4*x^4) + b^5*(-256*c^5 + 128*c^4*d*x - 96*c^3*d^2*x^2 + 80*c^2*d^3*x^3 - 70*c*d^4*
x^4 + 63*d^5*x^5)))/(693*d^6)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 45.97, size = 395, normalized size = 2.56 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (693 a^5 c d^5+693 a^5 d^6 x-2310 a^4 b c^2 d^4-1155 a^4 b c d^5 x+1155 a^4 b d^6 x^2+3696 a^3 b^2 c^3 d^3+1848 a^3 b^2 c^2 d^4 x-462 a^3 b^2 c d^5 x^2+1386 a^3 b^2 d^6 x^3-3168 a^2 b^3 c^4 d^2-1584 a^2 b^3 c^3 d^3 x+396 a^2 b^3 c^2 d^4 x^2-198 a^2 b^3 c d^5 x^3+990 a^2 b^3 d^6 x^4+1408 a b^4 c^5 d+704 a b^4 c^4 d^2 x-176 a b^4 c^3 d^3 x^2+88 a b^4 c^2 d^4 x^3-55 a b^4 c d^5 x^4+385 a b^4 d^6 x^5-256 b^5 c^6-128 b^5 c^5 d x+32 b^5 c^4 d^2 x^2-16 b^5 c^3 d^3 x^3+10 b^5 c^2 d^4 x^4-7 b^5 c d^5 x^5+63 b^5 d^6 x^6\right )}{693 d^6 \sqrt {c+d x}},d\text {!=}0\right \}\right \},\frac {\text {Piecewise}\left [\left \{\left \{a^5 x,b\text {==}0\right \},\left \{\frac {\left (a+b x\right )^6}{6 b},\text {True}\right \}\right \}\right ]}{\sqrt {c}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^5/(c + d*x)^(1/2),x]')

[Out]

Piecewise[{{2 (693 a ^ 5 c d ^ 5 + 693 a ^ 5 d ^ 6 x - 2310 a ^ 4 b c ^ 2 d ^ 4 - 1155 a ^ 4 b c d ^ 5 x + 115
5 a ^ 4 b d ^ 6 x ^ 2 + 3696 a ^ 3 b ^ 2 c ^ 3 d ^ 3 + 1848 a ^ 3 b ^ 2 c ^ 2 d ^ 4 x - 462 a ^ 3 b ^ 2 c d ^
5 x ^ 2 + 1386 a ^ 3 b ^ 2 d ^ 6 x ^ 3 - 3168 a ^ 2 b ^ 3 c ^ 4 d ^ 2 - 1584 a ^ 2 b ^ 3 c ^ 3 d ^ 3 x + 396 a
^ 2 b ^ 3 c ^ 2 d ^ 4 x ^ 2 - 198 a ^ 2 b ^ 3 c d ^ 5 x ^ 3 + 990 a ^ 2 b ^ 3 d ^ 6 x ^ 4 + 1408 a b ^ 4 c ^
5 d + 704 a b ^ 4 c ^ 4 d ^ 2 x - 176 a b ^ 4 c ^ 3 d ^ 3 x ^ 2 + 88 a b ^ 4 c ^ 2 d ^ 4 x ^ 3 - 55 a b ^ 4 c
d ^ 5 x ^ 4 + 385 a b ^ 4 d ^ 6 x ^ 5 - 256 b ^ 5 c ^ 6 - 128 b ^ 5 c ^ 5 d x + 32 b ^ 5 c ^ 4 d ^ 2 x ^ 2 - 1
6 b ^ 5 c ^ 3 d ^ 3 x ^ 3 + 10 b ^ 5 c ^ 2 d ^ 4 x ^ 4 - 7 b ^ 5 c d ^ 5 x ^ 5 + 63 b ^ 5 d ^ 6 x ^ 6) / (693
d ^ 6 Sqrt[c + d x]), d != 0}}, Piecewise[{{a ^ 5 x, b == 0}, {(a + b x) ^ 6 / (6 b), True}}] / Sqrt[c]]

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Maple [A]
time = 0.15, size = 121, normalized size = 0.79

method	result	size
derivativedivides	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+4 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{5} \sqrt {d x +c}}{d^{6}}\)	\(121\)
default	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+4 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{5} \sqrt {d x +c}}{d^{6}}\)	\(121\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (63 b^{5} x^{5} d^{5}+385 a \,b^{4} d^{5} x^{4}-70 b^{5} c \,d^{4} x^{4}+990 a^{2} b^{3} d^{5} x^{3}-440 a \,b^{4} c \,d^{4} x^{3}+80 b^{5} c^{2} d^{3} x^{3}+1386 a^{3} b^{2} d^{5} x^{2}-1188 a^{2} b^{3} c \,d^{4} x^{2}+528 a \,b^{4} c^{2} d^{3} x^{2}-96 b^{5} c^{3} d^{2} x^{2}+1155 a^{4} b \,d^{5} x -1848 a^{3} b^{2} c \,d^{4} x +1584 a^{2} b^{3} c^{2} d^{3} x -704 a \,b^{4} c^{3} d^{2} x +128 b^{5} c^{4} d x +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\)	\(273\)
trager	\(\frac {2 \sqrt {d x +c}\, \left (63 b^{5} x^{5} d^{5}+385 a \,b^{4} d^{5} x^{4}-70 b^{5} c \,d^{4} x^{4}+990 a^{2} b^{3} d^{5} x^{3}-440 a \,b^{4} c \,d^{4} x^{3}+80 b^{5} c^{2} d^{3} x^{3}+1386 a^{3} b^{2} d^{5} x^{2}-1188 a^{2} b^{3} c \,d^{4} x^{2}+528 a \,b^{4} c^{2} d^{3} x^{2}-96 b^{5} c^{3} d^{2} x^{2}+1155 a^{4} b \,d^{5} x -1848 a^{3} b^{2} c \,d^{4} x +1584 a^{2} b^{3} c^{2} d^{3} x -704 a \,b^{4} c^{3} d^{2} x +128 b^{5} c^{4} d x +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\)	\(273\)
risch	\(\frac {2 \sqrt {d x +c}\, \left (63 b^{5} x^{5} d^{5}+385 a \,b^{4} d^{5} x^{4}-70 b^{5} c \,d^{4} x^{4}+990 a^{2} b^{3} d^{5} x^{3}-440 a \,b^{4} c \,d^{4} x^{3}+80 b^{5} c^{2} d^{3} x^{3}+1386 a^{3} b^{2} d^{5} x^{2}-1188 a^{2} b^{3} c \,d^{4} x^{2}+528 a \,b^{4} c^{2} d^{3} x^{2}-96 b^{5} c^{3} d^{2} x^{2}+1155 a^{4} b \,d^{5} x -1848 a^{3} b^{2} c \,d^{4} x +1584 a^{2} b^{3} c^{2} d^{3} x -704 a \,b^{4} c^{3} d^{2} x +128 b^{5} c^{4} d x +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d^6*(1/11*b^5*(d*x+c)^(11/2)+5/9*(a*d-b*c)*b^4*(d*x+c)^(9/2)+10/7*(a*d-b*c)^2*b^3*(d*x+c)^(7/2)+2*(a*d-b*c)^
3*b^2*(d*x+c)^(5/2)+5/3*(a*d-b*c)^4*b*(d*x+c)^(3/2)+(a*d-b*c)^5*(d*x+c)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
time = 0.27, size = 283, normalized size = 1.84 \begin {gather*} \frac {2 \, {\left (693 \, \sqrt {d x + c} a^{5} + \frac {1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} b}{d} + \frac {462 \, {\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^{2}}{d^{2}} + \frac {198 \, {\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^{3}}{d^{3}} + \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 )} a b^{4}}{d^{4}} + \frac {{\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^{5}}{d^{5}}\right )}}{693 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b/d + 462*(3*(d*x + c)^(5/2) - 1
0*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2/d^2 + 198*(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^3/d^3 + 11*(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^4/d^4 + (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^5/d^5)/d

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Fricas [A]
time = 0.29, size = 261, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 1408 \, a b^{4} c^{4} d - 3168 \, a^{2} b^{3} c^{3} d^{2} + 3696 \, a^{3} b^{2} c^{2} d^{3} - 2310 \, a^{4} b c d^{4} + 693 \, a^{5} d^{5} - 35 \, {\left (2 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} c^{2} d^{3} - 44 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} c^{3} d^{2} - 88 \, a b^{4} c^{2} d^{3} + 198 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} + {\left (128 \, b^{5} c^{4} d - 704 \, a b^{4} c^{3} d^{2} + 1584 \, a^{2} b^{3} c^{2} d^{3} - 1848 \, a^{3} b^{2} c d^{4} + 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{693 \, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/693*(63*b^5*d^5*x^5 - 256*b^5*c^5 + 1408*a*b^4*c^4*d - 3168*a^2*b^3*c^3*d^2 + 3696*a^3*b^2*c^2*d^3 - 2310*a^
4*b*c*d^4 + 693*a^5*d^5 - 35*(2*b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 10*(8*b^5*c^2*d^3 - 44*a*b^4*c*d^4 + 99*a^2*b^
3*d^5)*x^3 - 6*(16*b^5*c^3*d^2 - 88*a*b^4*c^2*d^3 + 198*a^2*b^3*c*d^4 - 231*a^3*b^2*d^5)*x^2 + (128*b^5*c^4*d
- 704*a*b^4*c^3*d^2 + 1584*a^2*b^3*c^2*d^3 - 1848*a^3*b^2*c*d^4 + 1155*a^4*b*d^5)*x)*sqrt(d*x + c)/d^6

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Sympy [A]
time = 36.74, size = 728, normalized size = 4.73 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{5} c}{\sqrt {c + d x}} - 2 a^{5} \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {10 a^{4} b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {10 a^{4} 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 {20 a^{3} 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 {20 a^{3} 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 {20 a^{2} 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 {20 a^{2} 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 {10 a 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 {10 a 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}} - \frac {2 b^{5} c \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^{5}} - \frac {2 b^{5} \left (\frac {c^{6}}{\sqrt {c + d x}} + 6 c^{5} \sqrt {c + d x} - 5 c^{4} \left (c + d x\right )^{\frac {3}{2}} + 4 c^{3} \left (c + d x\right )^{\frac {5}{2}} - \frac {15 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {2 c \left (c + d x\right )^{\frac {9}{2}}}{3} - \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{5}}}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 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)**5/(d*x+c)**(1/2),x)

[Out]

Piecewise(((-2*a**5*c/sqrt(c + d*x) - 2*a**5*(-c/sqrt(c + d*x) - sqrt(c + d*x)) - 10*a**4*b*c*(-c/sqrt(c + d*x
) - sqrt(c + d*x))/d - 10*a**4*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d - 20*a**3*b**
2*c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 - 20*a**3*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 - 20*a**2*b**3*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 - 20*a**2*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 - 10*a*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 - 10*a*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 - 2*b**5*c*(-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*
*5 - 2*b**5*(c**6/sqrt(c + d*x) + 6*c**5*sqrt(c + d*x) - 5*c**4*(c + d*x)**(3/2) + 4*c**3*(c + d*x)**(5/2) - 1
5*c**2*(c + d*x)**(7/2)/7 + 2*c*(c + d*x)**(9/2)/3 - (c + d*x)**(11/2)/11)/d**5)/d, Ne(d, 0)), (Piecewise((a**
5*x, Eq(b, 0)), ((a + b*x)**6/(6*b), True))/sqrt(c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
time = 0.00, size = 448, normalized size = 2.91 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{11} \sqrt {c+d x} \left (c+d x\right )^{5}-\frac {5}{9} \sqrt {c+d x} \left (c+d x\right )^{4} c+\frac {10}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c^{2}-2 \sqrt {c+d x} \left (c+d x\right )^{2} c^{3}+\frac {5}{3} \sqrt {c+d x} \left (c+d x\right ) c^{4}-\sqrt {c+d x} c^{5}\right )}{d^{5}}+\frac {10 a b^{4} \left (\frac {1}{9} \sqrt {c+d x} \left (c+d x\right )^{4}-\frac {4}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c+\frac {6}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c^{2}-\frac {4}{3} \sqrt {c+d x} \left (c+d x\right ) c^{3}+\sqrt {c+d x} c^{4}\right )}{d^{4}}+\frac {20 a^{2} b^{3} \left (\frac {1}{7} \sqrt {c+d x} \left (c+d x\right )^{3}-\frac {3}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c+\sqrt {c+d x} \left (c+d x\right ) c^{2}-\sqrt {c+d x} c^{3}\right )}{d^{3}}+\frac {20 a^{3} b^{2} \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+\frac {10 a^{4} b \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a^{5} \sqrt {c+d x}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^(1/2),x)

[Out]

2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b/d + 462*(3*(d*x + c)^(5/2) - 1
0*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2/d^2 + 198*(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^3/d^3 + 11*(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^4/d^4 + (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^5/d^5)/d

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Mupad [B]
time = 0.07, size = 137, normalized size = 0.89 \begin {gather*} \frac {2\,b^5\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,\sqrt {c+d\,x}}{d^6}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(c + d*x)^(1/2),x)

[Out]

(2*b^5*(c + d*x)^(11/2))/(11*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(9/2))/(9*d^6) + (2*(a*d - b*c)^5*(c +
d*x)^(1/2))/d^6 + (4*b^2*(a*d - b*c)^3*(c + d*x)^(5/2))/d^6 + (20*b^3*(a*d - b*c)^2*(c + d*x)^(7/2))/(7*d^6) +
(10*b*(a*d - b*c)^4*(c + d*x)^(3/2))/(3*d^6)

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