3.14.100 \(\int (a+b x)^4 (c+d x)^{5/2} \, dx\) [1400]

Optimal. Leaf size=129 \[ \frac {2 (b c-a d)^4 (c+d x)^{7/2}}{7 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{9/2}}{9 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{11/2}}{11 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{13/2}}{13 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5} \]

[Out]

2/7*(-a*d+b*c)^4*(d*x+c)^(7/2)/d^5-8/9*b*(-a*d+b*c)^3*(d*x+c)^(9/2)/d^5+12/11*b^2*(-a*d+b*c)^2*(d*x+c)^(11/2)/
d^5-8/13*b^3*(-a*d+b*c)*(d*x+c)^(13/2)/d^5+2/15*b^4*(d*x+c)^(15/2)/d^5

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Rubi [A]
time = 0.03, 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)^{13/2} (b c-a d)}{13 d^5}+\frac {12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac {8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac {2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/(9*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*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 (c+d x)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^4 (c+d x)^{5/2}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{7/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{9/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{11/2}}{d^4}+\frac {b^4 (c+d x)^{13/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 (c+d x)^{7/2}}{7 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{9/2}}{9 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{11/2}}{11 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{13/2}}{13 d^5}+\frac {2 b^4 (c+d x)^{15/2}}{15 d^5}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 154, normalized size = 1.19 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (6435 a^4 d^4+2860 a^3 b d^3 (-2 c+7 d x)+390 a^2 b^2 d^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )+60 a b^3 d \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+b^4 \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )\right )}{45045 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(6435*a^4*d^4 + 2860*a^3*b*d^3*(-2*c + 7*d*x) + 390*a^2*b^2*d^2*(8*c^2 - 28*c*d*x + 63*d^2*
x^2) + 60*a*b^3*d*(-16*c^3 + 56*c^2*d*x - 126*c*d^2*x^2 + 231*d^3*x^3) + b^4*(128*c^4 - 448*c^3*d*x + 1008*c^2
*d^2*x^2 - 1848*c*d^3*x^3 + 3003*d^4*x^4)))/(45045*d^5)

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Maple [A]
time = 0.15, size = 100, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\)	\(100\)
default	\(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\)	\(100\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (3003 d^{4} x^{4} b^{4}+13860 a \,b^{3} d^{4} x^{3}-1848 b^{4} c \,d^{3} x^{3}+24570 a^{2} b^{2} d^{4} x^{2}-7560 a \,b^{3} c \,d^{3} x^{2}+1008 b^{4} c^{2} d^{2} x^{2}+20020 a^{3} b \,d^{4} x -10920 a^{2} b^{2} c \,d^{3} x +3360 a \,b^{3} c^{2} d^{2} x -448 b^{4} c^{3} d x +6435 a^{4} d^{4}-5720 a^{3} b c \,d^{3}+3120 a^{2} b^{2} c^{2} d^{2}-960 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{45045 d^{5}}\)	\(186\)
trager	\(\frac {2 \left (3003 b^{4} d^{7} x^{7}+13860 a \,b^{3} d^{7} x^{6}+7161 b^{4} c \,d^{6} x^{6}+24570 b^{2} a^{2} d^{7} x^{5}+34020 a \,b^{3} c \,d^{6} x^{5}+4473 b^{4} c^{2} d^{5} x^{5}+20020 a^{3} b \,d^{7} x^{4}+62790 b^{2} a^{2} c \,d^{6} x^{4}+22260 a \,b^{3} c^{2} d^{5} x^{4}+35 b^{4} c^{3} d^{4} x^{4}+6435 a^{4} d^{7} x^{3}+54340 a^{3} b c \,d^{6} x^{3}+44070 b^{2} a^{2} c^{2} d^{5} x^{3}+300 a \,b^{3} c^{3} d^{4} x^{3}-40 b^{4} c^{4} d^{3} x^{3}+19305 a^{4} c \,d^{6} x^{2}+42900 a^{3} b \,c^{2} d^{5} x^{2}+1170 a^{2} b^{2} c^{3} d^{4} x^{2}-360 a \,b^{3} c^{4} d^{3} x^{2}+48 b^{4} c^{5} d^{2} x^{2}+19305 a^{4} c^{2} d^{5} x +2860 a^{3} b \,c^{3} d^{4} x -1560 b^{2} a^{2} c^{4} d^{3} x +480 a \,b^{3} c^{5} d^{2} x -64 b^{4} c^{6} d x +6435 a^{4} c^{3} d^{4}-5720 a^{3} b \,c^{4} d^{3}+3120 b^{2} a^{2} c^{5} d^{2}-960 a \,b^{3} c^{6} d +128 b^{4} c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\)	\(407\)
risch	\(\frac {2 \left (3003 b^{4} d^{7} x^{7}+13860 a \,b^{3} d^{7} x^{6}+7161 b^{4} c \,d^{6} x^{6}+24570 b^{2} a^{2} d^{7} x^{5}+34020 a \,b^{3} c \,d^{6} x^{5}+4473 b^{4} c^{2} d^{5} x^{5}+20020 a^{3} b \,d^{7} x^{4}+62790 b^{2} a^{2} c \,d^{6} x^{4}+22260 a \,b^{3} c^{2} d^{5} x^{4}+35 b^{4} c^{3} d^{4} x^{4}+6435 a^{4} d^{7} x^{3}+54340 a^{3} b c \,d^{6} x^{3}+44070 b^{2} a^{2} c^{2} d^{5} x^{3}+300 a \,b^{3} c^{3} d^{4} x^{3}-40 b^{4} c^{4} d^{3} x^{3}+19305 a^{4} c \,d^{6} x^{2}+42900 a^{3} b \,c^{2} d^{5} x^{2}+1170 a^{2} b^{2} c^{3} d^{4} x^{2}-360 a \,b^{3} c^{4} d^{3} x^{2}+48 b^{4} c^{5} d^{2} x^{2}+19305 a^{4} c^{2} d^{5} x +2860 a^{3} b \,c^{3} d^{4} x -1560 b^{2} a^{2} c^{4} d^{3} x +480 a \,b^{3} c^{5} d^{2} x -64 b^{4} c^{6} d x +6435 a^{4} c^{3} d^{4}-5720 a^{3} b \,c^{4} d^{3}+3120 b^{2} a^{2} c^{5} d^{2}-960 a \,b^{3} c^{6} d +128 b^{4} c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\)	\(407\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d^5*(1/15*b^4*(d*x+c)^(15/2)+4/13*(a*d-b*c)*b^3*(d*x+c)^(13/2)+6/11*(a*d-b*c)^2*b^2*(d*x+c)^(11/2)+4/9*(a*d-
b*c)^3*b*(d*x+c)^(9/2)+1/7*(a*d-b*c)^4*(d*x+c)^(7/2))

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Maxima [A]
time = 0.32, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 24570 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 20020 \, {\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 {9}{2}} + 6435 \, {\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 {7}{2}}\right )}}{45045 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(d*x + c)^(15/2)*b^4 - 13860*(b^4*c - a*b^3*d)*(d*x + c)^(13/2) + 24570*(b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*(d*x + c)^(11/2) - 20020*(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)^(9/2)
+ 6435*(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)^(7/2))/d^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (109) = 218\).
time = 0.89, size = 377, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left (3003 \, b^{4} d^{7} x^{7} + 128 \, b^{4} c^{7} - 960 \, a b^{3} c^{6} d + 3120 \, a^{2} b^{2} c^{5} d^{2} - 5720 \, a^{3} b c^{4} d^{3} + 6435 \, a^{4} c^{3} d^{4} + 231 \, {\left (31 \, b^{4} c d^{6} + 60 \, a b^{3} d^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} c^{2} d^{5} + 540 \, a b^{3} c d^{6} + 390 \, a^{2} b^{2} d^{7}\right )} x^{5} + 35 \, {\left (b^{4} c^{3} d^{4} + 636 \, a b^{3} c^{2} d^{5} + 1794 \, a^{2} b^{2} c d^{6} + 572 \, a^{3} b d^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} c^{4} d^{3} - 60 \, a b^{3} c^{3} d^{4} - 8814 \, a^{2} b^{2} c^{2} d^{5} - 10868 \, a^{3} b c d^{6} - 1287 \, a^{4} d^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{5} d^{2} - 120 \, a b^{3} c^{4} d^{3} + 390 \, a^{2} b^{2} c^{3} d^{4} + 14300 \, a^{3} b c^{2} d^{5} + 6435 \, a^{4} c d^{6}\right )} x^{2} - {\left (64 \, b^{4} c^{6} d - 480 \, a b^{3} c^{5} d^{2} + 1560 \, a^{2} b^{2} c^{4} d^{3} - 2860 \, a^{3} b c^{3} d^{4} - 19305 \, a^{4} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*d^7*x^7 + 128*b^4*c^7 - 960*a*b^3*c^6*d + 3120*a^2*b^2*c^5*d^2 - 5720*a^3*b*c^4*d^3 + 6435*a
^4*c^3*d^4 + 231*(31*b^4*c*d^6 + 60*a*b^3*d^7)*x^6 + 63*(71*b^4*c^2*d^5 + 540*a*b^3*c*d^6 + 390*a^2*b^2*d^7)*x
^5 + 35*(b^4*c^3*d^4 + 636*a*b^3*c^2*d^5 + 1794*a^2*b^2*c*d^6 + 572*a^3*b*d^7)*x^4 - 5*(8*b^4*c^4*d^3 - 60*a*b
^3*c^3*d^4 - 8814*a^2*b^2*c^2*d^5 - 10868*a^3*b*c*d^6 - 1287*a^4*d^7)*x^3 + 3*(16*b^4*c^5*d^2 - 120*a*b^3*c^4*
d^3 + 390*a^2*b^2*c^3*d^4 + 14300*a^3*b*c^2*d^5 + 6435*a^4*c*d^6)*x^2 - (64*b^4*c^6*d - 480*a*b^3*c^5*d^2 + 15
60*a^2*b^2*c^4*d^3 - 2860*a^3*b*c^3*d^4 - 19305*a^4*c^2*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]
time = 16.76, size = 960, normalized size = 7.44 \begin {gather*} a^{4} c^{2} \left (\begin {cases} \sqrt {c} x & \text {for}\: d = 0 \\\frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + \frac {4 a^{4} c \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d} + \frac {2 a^{4} \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d} + \frac {8 a^{3} b c^{2} \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} + \frac {16 a^{3} b c \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{2}} + \frac {8 a^{3} b \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{2}} + \frac {12 a^{2} b^{2} c^{2} \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} + \frac {24 a^{2} b^{2} c \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} + \frac {12 a^{2} b^{2} \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{3}} + \frac {8 a b^{3} c^{2} \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} + \frac {16 a b^{3} c \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} + \frac {8 a b^{3} \left (- \frac {c^{5} \left (c + d x\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + d x\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + d x\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + d x\right )^{\frac {11}{2}}}{11} + \frac {\left (c + d x\right )^{\frac {13}{2}}}{13}\right )}{d^{4}} + \frac {2 b^{4} c^{2} \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} + \frac {4 b^{4} c \left (- \frac {c^{5} \left (c + d x\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + d x\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + d x\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + d x\right )^{\frac {11}{2}}}{11} + \frac {\left (c + d x\right )^{\frac {13}{2}}}{13}\right )}{d^{5}} + \frac {2 b^{4} \left (\frac {c^{6} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {6 c^{5} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {15 c^{4} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {20 c^{3} \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {15 c^{2} \left (c + d x\right )^{\frac {11}{2}}}{11} - \frac {6 c \left (c + d x\right )^{\frac {13}{2}}}{13} + \frac {\left (c + d x\right )^{\frac {15}{2}}}{15}\right )}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c)**(5/2),x)

[Out]

a**4*c**2*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 4*a**4*c*(-c*(c + d*x)**(3/2)/3
+ (c + d*x)**(5/2)/5)/d + 2*a**4*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d +
8*a**3*b*c**2*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 16*a**3*b*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*(
c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 8*a**3*b*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**2 + 12*a**2*b**2*c**2*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*
x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 24*a**2*b**2*c*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**3 + 12*a**2*b**2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)
**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**3 + 8*a*b**3*c**2*(-
c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + 16*a
*b**3*c*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2
)/9 + (c + d*x)**(11/2)/11)/d**4 + 8*a*b**3*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d
*x)**(7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**4 + 2*b**4*c**
2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 +
(c + d*x)**(11/2)/11)/d**5 + 4*b**4*c*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d*x)**(
7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**5 + 2*b**4*(c**6*(c
+ d*x)**(3/2)/3 - 6*c**5*(c + d*x)**(5/2)/5 + 15*c**4*(c + d*x)**(7/2)/7 - 20*c**3*(c + d*x)**(9/2)/9 + 15*c**
2*(c + d*x)**(11/2)/11 - 6*c*(c + d*x)**(13/2)/13 + (c + d*x)**(15/2)/15)/d**5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (109) = 218\).
time = 1.79, size = 1204, normalized size = 9.33 \begin {gather*} \frac {2 \, {\left (45045 \, \sqrt {d x + c} a^{4} c^{3} + 45045 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} c^{2} + \frac {60060 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b c^{3}}{d} + 9009 \, {\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^{4} c + \frac {18018 \, {\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^{3}}{d^{2}} + \frac {36036 \, {\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 c^{2}}{d} + 1287 \, {\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^{4} + \frac {5148 \, {\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^{3}}{d^{3}} + \frac {23166 \, {\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} c^{2}}{d^{2}} + \frac {15444 \, {\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^{3} b c}{d} + \frac {143 \, {\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^{3}}{d^{4}} + \frac {1716 \, {\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} c^{2}}{d^{3}} + \frac {2574 \, {\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^{2} b^{2} c}{d^{2}} + \frac {572 \, {\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^{3} b}{d} + \frac {195 \, {\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} c^{2}}{d^{4}} + \frac {780 \, {\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 )} a b^{3} c}{d^{3}} + \frac {390 \, {\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 )} a^{2} b^{2}}{d^{2}} + \frac {45 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} - 1638 \, {\left (d x + c\right )}^{\frac {11}{2}} c + 5005 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{2} - 8580 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{3} + 9009 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{4} - 6006 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{5} + 3003 \, \sqrt {d x + c} c^{6}\right )} b^{4} c}{d^{4}} + \frac {60 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} - 1638 \, {\left (d x + c\right )}^{\frac {11}{2}} c + 5005 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{2} - 8580 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{3} + 9009 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{4} - 6006 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{5} + 3003 \, \sqrt {d x + c} c^{6}\right )} a b^{3}}{d^{3}} + \frac {7 \, {\left (429 \, {\left (d x + c\right )}^{\frac {15}{2}} - 3465 \, {\left (d x + c\right )}^{\frac {13}{2}} c + 12285 \, {\left (d x + c\right )}^{\frac {11}{2}} c^{2} - 25025 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{3} + 32175 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{4} - 27027 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{5} + 15015 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{6} - 6435 \, \sqrt {d x + c} c^{7}\right )} b^{4}}{d^{4}}\right )}}{45045 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(d*x + c)*a^4*c^3 + 45045*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*c^2 + 60060*((d*x + c)^
(3/2) - 3*sqrt(d*x + c)*c)*a^3*b*c^3/d + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2
)*a^4*c + 18018*(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^3/d^2 + 36036*(3*(
d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b*c^2/d + 1287*(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^4 + 5148*(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^3/d^3 + 23166*(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*c^2/d^2 + 15444*(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^3*b*c/d + 143*(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*c^3/d^4 + 1716*(3
5*(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*c^2/d^3 + 2574*(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^2*b^2*c/d^2 + 572*(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^3*b/d + 195*(63*(d*x + c)^(11/2) - 3
85*(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*sqr
t(d*x + c)*c^5)*b^4*c^2/d^4 + 780*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 138
6*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*a*b^3*c/d^3 + 390*(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 - 6
93*sqrt(d*x + c)*c^5)*a^2*b^2/d^2 + 45*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*
c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)
*b^4*c/d^4 + 60*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7
/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*a*b^3/d^3 + 7*(429*(d*
x + c)^(15/2) - 3465*(d*x + c)^(13/2)*c + 12285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175*(d*x
+ c)^(7/2)*c^4 - 27027*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6 - 6435*sqrt(d*x + c)*c^7)*b^4/d^4)/d

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Mupad [B]
time = 0.23, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (c+d\,x\right )}^{15/2}}{15\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4*(c + d*x)^(5/2),x)

[Out]

(2*b^4*(c + d*x)^(15/2))/(15*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(13/2))/(13*d^5) + (2*(a*d - b*c)^4*(c +
d*x)^(7/2))/(7*d^5) + (12*b^2*(a*d - b*c)^2*(c + d*x)^(11/2))/(11*d^5) + (8*b*(a*d - b*c)^3*(c + d*x)^(9/2))/(
9*d^5)

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