3.15.1 \(\int (a+b x)^3 (c+d x)^{5/2} \, dx\) [1401]

Optimal. Leaf size=100 \[ -\frac {2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{9/2}}{3 d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{11/2}}{11 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 100, 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 {6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac {2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac {2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^(5/2),x]

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 102, normalized size = 1.02 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (-2 c+7 d x)+13 a b^2 d \left (8 c^2-28 c d x+63 d^2 x^2\right )+b^3 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )\right )}{3003 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(429*a^3*d^3 + 143*a^2*b*d^2*(-2*c + 7*d*x) + 13*a*b^2*d*(8*c^2 - 28*c*d*x + 63*d^2*x^2) +
b^3*(-16*c^3 + 56*c^2*d*x - 126*c*d^2*x^2 + 231*d^3*x^3)))/(3003*d^4)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 4.60, size = 276, normalized size = 2.76 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-16 b^3 c^6+8 b^2 c^5 d \left (13 a+b x\right )-2 b c^4 d^2 \left (143 a^2+26 a b x+3 b^2 x^2\right )+c^3 d^3 \left (429 a^3+143 a^2 b x+39 a b^2 x^2+5 b^3 x^3\right )+d^4 x \left (1287 a^3 c^2+1287 a^3 c d x+429 a^3 d^2 x^2+2145 a^2 b c^2 x+2717 a^2 b c d x^2+1001 a^2 b d^2 x^3+1469 a b^2 c^2 x^2+2093 a b^2 c d x^3+819 a b^2 d^2 x^4+371 b^3 c^2 x^3+567 b^3 c d x^4+231 b^3 d^2 x^5\right )\right ) \sqrt {c+d x}}{3003 d^4},d\text {!=}0\right \}\right \},c^{\frac {5}{2}} \left (a^3 x+\frac {3 a^2 b x^2}{2}+a b^2 x^3+\frac {b^3 x^4}{4}\right )\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^3*(c + d*x)^(5/2),x]')

[Out]

Piecewise[{{2 (-16 b ^ 3 c ^ 6 + 8 b ^ 2 c ^ 5 d (13 a + b x) - 2 b c ^ 4 d ^ 2 (143 a ^ 2 + 26 a b x + 3 b ^
2 x ^ 2) + c ^ 3 d ^ 3 (429 a ^ 3 + 143 a ^ 2 b x + 39 a b ^ 2 x ^ 2 + 5 b ^ 3 x ^ 3) + d ^ 4 x (1287 a ^ 3 c
^ 2 + 1287 a ^ 3 c d x + 429 a ^ 3 d ^ 2 x ^ 2 + 2145 a ^ 2 b c ^ 2 x + 2717 a ^ 2 b c d x ^ 2 + 1001 a ^ 2 b
d ^ 2 x ^ 3 + 1469 a b ^ 2 c ^ 2 x ^ 2 + 2093 a b ^ 2 c d x ^ 3 + 819 a b ^ 2 d ^ 2 x ^ 4 + 371 b ^ 3 c ^ 2 x
^ 3 + 567 b ^ 3 c d x ^ 4 + 231 b ^ 3 d ^ 2 x ^ 5)) Sqrt[c + d x] / (3003 d ^ 4), d != 0}}, c ^ (5 / 2) (a ^ 3
x + 3 a ^ 2 b x ^ 2 / 2 + a b ^ 2 x ^ 3 + b ^ 3 x ^ 4 / 4)]

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Maple [A]
time = 0.14, size = 78, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\)	\(78\)
default	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\)	\(78\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (231 b^{3} x^{3} d^{3}+819 a \,b^{2} d^{3} x^{2}-126 b^{3} c \,d^{2} x^{2}+1001 a^{2} b \,d^{3} x -364 a \,b^{2} c \,d^{2} x +56 b^{3} c^{2} d x +429 a^{3} d^{3}-286 a^{2} b c \,d^{2}+104 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{3003 d^{4}}\)	\(116\)
trager	\(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {d x +c}}{3003 d^{4}}\)	\(286\)
risch	\(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {d x +c}}{3003 d^{4}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{3003 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*(d*x + c)^(13/2)*b^3 - 819*(b^3*c - a*b^2*d)*(d*x + c)^(11/2) + 1001*(b^3*c^2 - 2*a*b^2*c*d + a^2*
b*d^2)*(d*x + c)^(9/2) - 429*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(7/2))/d^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (84) = 168\).
time = 0.29, size = 268, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left (231 \, b^{3} d^{6} x^{6} - 16 \, b^{3} c^{6} + 104 \, a b^{2} c^{5} d - 286 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} + 63 \, {\left (9 \, b^{3} c d^{5} + 13 \, a b^{2} d^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} c^{2} d^{4} + 299 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} + {\left (5 \, b^{3} c^{3} d^{3} + 1469 \, a b^{2} c^{2} d^{4} + 2717 \, a^{2} b c d^{5} + 429 \, a^{3} d^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} c^{4} d^{2} - 13 \, a b^{2} c^{3} d^{3} - 715 \, a^{2} b c^{2} d^{4} - 429 \, a^{3} c d^{5}\right )} x^{2} + {\left (8 \, b^{3} c^{5} d - 52 \, a b^{2} c^{4} d^{2} + 143 \, a^{2} b c^{3} d^{3} + 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{3003 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^3*d^6*x^6 - 16*b^3*c^6 + 104*a*b^2*c^5*d - 286*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 + 63*(9*b^3*c*d^5
+ 13*a*b^2*d^6)*x^5 + 7*(53*b^3*c^2*d^4 + 299*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 + (5*b^3*c^3*d^3 + 1469*a*b^2*
c^2*d^4 + 2717*a^2*b*c*d^5 + 429*a^3*d^6)*x^3 - 3*(2*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 - 715*a^2*b*c^2*d^4 - 429*
a^3*c*d^5)*x^2 + (8*b^3*c^5*d - 52*a*b^2*c^4*d^2 + 143*a^2*b*c^3*d^3 + 1287*a^3*c^2*d^4)*x)*sqrt(d*x + c)/d^4

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Sympy [A]
time = 0.45, size = 549, normalized size = 5.49 \begin {gather*} \begin {cases} \frac {2 a^{3} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{3} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{3} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{3} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {4 a^{2} b c^{4} \sqrt {c + d x}}{21 d^{2}} + \frac {2 a^{2} b c^{3} x \sqrt {c + d x}}{21 d} + \frac {10 a^{2} b c^{2} x^{2} \sqrt {c + d x}}{7} + \frac {38 a^{2} b c d x^{3} \sqrt {c + d x}}{21} + \frac {2 a^{2} b d^{2} x^{4} \sqrt {c + d x}}{3} + \frac {16 a b^{2} c^{5} \sqrt {c + d x}}{231 d^{3}} - \frac {8 a b^{2} c^{4} x \sqrt {c + d x}}{231 d^{2}} + \frac {2 a b^{2} c^{3} x^{2} \sqrt {c + d x}}{77 d} + \frac {226 a b^{2} c^{2} x^{3} \sqrt {c + d x}}{231} + \frac {46 a b^{2} c d x^{4} \sqrt {c + d x}}{33} + \frac {6 a b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} - \frac {32 b^{3} c^{6} \sqrt {c + d x}}{3003 d^{4}} + \frac {16 b^{3} c^{5} x \sqrt {c + d x}}{3003 d^{3}} - \frac {4 b^{3} c^{4} x^{2} \sqrt {c + d x}}{1001 d^{2}} + \frac {10 b^{3} c^{3} x^{3} \sqrt {c + d x}}{3003 d} + \frac {106 b^{3} c^{2} x^{4} \sqrt {c + d x}}{429} + \frac {54 b^{3} c d x^{5} \sqrt {c + d x}}{143} + \frac {2 b^{3} d^{2} x^{6} \sqrt {c + d x}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a**3*c**3*sqrt(c + d*x)/(7*d) + 6*a**3*c**2*x*sqrt(c + d*x)/7 + 6*a**3*c*d*x**2*sqrt(c + d*x)/7 +
2*a**3*d**2*x**3*sqrt(c + d*x)/7 - 4*a**2*b*c**4*sqrt(c + d*x)/(21*d**2) + 2*a**2*b*c**3*x*sqrt(c + d*x)/(21*
d) + 10*a**2*b*c**2*x**2*sqrt(c + d*x)/7 + 38*a**2*b*c*d*x**3*sqrt(c + d*x)/21 + 2*a**2*b*d**2*x**4*sqrt(c + d
*x)/3 + 16*a*b**2*c**5*sqrt(c + d*x)/(231*d**3) - 8*a*b**2*c**4*x*sqrt(c + d*x)/(231*d**2) + 2*a*b**2*c**3*x**
2*sqrt(c + d*x)/(77*d) + 226*a*b**2*c**2*x**3*sqrt(c + d*x)/231 + 46*a*b**2*c*d*x**4*sqrt(c + d*x)/33 + 6*a*b*
*2*d**2*x**5*sqrt(c + d*x)/11 - 32*b**3*c**6*sqrt(c + d*x)/(3003*d**4) + 16*b**3*c**5*x*sqrt(c + d*x)/(3003*d*
*3) - 4*b**3*c**4*x**2*sqrt(c + d*x)/(1001*d**2) + 10*b**3*c**3*x**3*sqrt(c + d*x)/(3003*d) + 106*b**3*c**2*x*
*4*sqrt(c + d*x)/429 + 54*b**3*c*d*x**5*sqrt(c + d*x)/143 + 2*b**3*d**2*x**6*sqrt(c + d*x)/13, Ne(d, 0)), (c**
(5/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (84) = 168\).
time = 0.01, size = 1412, normalized size = 14.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(5/2),x)

[Out]

2/15015*(15015*sqrt(d*x + c)*a^3*c^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3*c^2 + 15015*((d*x + c)^
(3/2) - 3*sqrt(d*x + c)*c)*a^2*b*c^3/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2
)*a^3*c + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b^2*c^3/d^2 + 9009*(3*(d*x
+ c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b*c^2/d + 429*(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 + 429*(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)*b^3*c^3/d^3 + 3861*(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^2*c^2/d^2 + 3861*(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*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^3*c^2/d^3 + 429*(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^
2*c/d^2 + 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)*a^2*b/d + 65*(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^3*c/d^3 + 65*(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)*a*b^2/d^2 + 5*(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
^3/d^3)/d

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Mupad [B]
time = 0.08, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (c+d\,x\right )}^{13/2}}{13\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3*(c + d*x)^(5/2),x)

[Out]

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

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