3.14.7 \(\int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx\)

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

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Rubi [A]  time = 0.05, 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 = {43} \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 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)^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 = 0.09, size = 123, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {c+d x} \left (-385 b^4 (c+d x)^4 (b c-a d)+990 b^3 (c+d x)^3 (b c-a d)^2-1386 b^2 (c+d x)^2 (b c-a d)^3+1155 b (c+d x) (b c-a d)^4-693 (b c-a d)^5+63 b^5 (c+d x)^5\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*(b*c - a*d)^5 + 1155*b*(b*c - a*d)^4*(c + d*x) - 1386*b^2*(b*c - a*d)^3*(c + d*x)^2 + 9
90*b^3*(b*c - a*d)^2*(c + d*x)^3 - 385*b^4*(b*c - a*d)*(c + d*x)^4 + 63*b^5*(c + d*x)^5))/(693*d^6)

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IntegrateAlgebraic [B]  time = 0.10, size = 315, normalized size = 2.05 \begin {gather*} \frac {2 \sqrt {c+d x} \left (693 a^5 d^5+1155 a^4 b d^4 (c+d x)-3465 a^4 b c d^4+6930 a^3 b^2 c^2 d^3+1386 a^3 b^2 d^3 (c+d x)^2-4620 a^3 b^2 c d^3 (c+d x)-6930 a^2 b^3 c^3 d^2+6930 a^2 b^3 c^2 d^2 (c+d x)+990 a^2 b^3 d^2 (c+d x)^3-4158 a^2 b^3 c d^2 (c+d x)^2+3465 a b^4 c^4 d-4620 a b^4 c^3 d (c+d x)+4158 a b^4 c^2 d (c+d x)^2+385 a b^4 d (c+d x)^4-1980 a b^4 c d (c+d x)^3-693 b^5 c^5+1155 b^5 c^4 (c+d x)-1386 b^5 c^3 (c+d x)^2+990 b^5 c^2 (c+d x)^3+63 b^5 (c+d x)^5-385 b^5 c (c+d x)^4\right )}{693 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-693*b^5*c^5 + 3465*a*b^4*c^4*d - 6930*a^2*b^3*c^3*d^2 + 6930*a^3*b^2*c^2*d^3 - 3465*a^4*b*c
*d^4 + 693*a^5*d^5 + 1155*b^5*c^4*(c + d*x) - 4620*a*b^4*c^3*d*(c + d*x) + 6930*a^2*b^3*c^2*d^2*(c + d*x) - 46
20*a^3*b^2*c*d^3*(c + d*x) + 1155*a^4*b*d^4*(c + d*x) - 1386*b^5*c^3*(c + d*x)^2 + 4158*a*b^4*c^2*d*(c + d*x)^
2 - 4158*a^2*b^3*c*d^2*(c + d*x)^2 + 1386*a^3*b^2*d^3*(c + d*x)^2 + 990*b^5*c^2*(c + d*x)^3 - 1980*a*b^4*c*d*(
c + d*x)^3 + 990*a^2*b^3*d^2*(c + d*x)^3 - 385*b^5*c*(c + d*x)^4 + 385*a*b^4*d*(c + d*x)^4 + 63*b^5*(c + d*x)^
5))/(693*d^6)

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fricas [A]  time = 1.26, 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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giac [B]  time = 1.07, 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="giac")

[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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maple [B]  time = 0.01, size = 273, normalized size = 1.77 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(d*x+c)^(1/2)*(63*b^5*d^5*x^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)/d^6

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maxima [B]  time = 1.38, 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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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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sympy [A]  time = 79.91, 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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