3.4.10 \(\int x^6 (a+b x)^{9/2} \, dx\) [310]

Optimal. Leaf size=127 \[ \frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7} \]

[Out]

2/11*a^6*(b*x+a)^(11/2)/b^7-12/13*a^5*(b*x+a)^(13/2)/b^7+2*a^4*(b*x+a)^(15/2)/b^7-40/17*a^3*(b*x+a)^(17/2)/b^7
+30/19*a^2*(b*x+a)^(19/2)/b^7-4/7*a*(b*x+a)^(21/2)/b^7+2/23*(b*x+a)^(23/2)/b^7

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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \begin {gather*} \frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^(9/2),x]

[Out]

(2*a^6*(a + b*x)^(11/2))/(11*b^7) - (12*a^5*(a + b*x)^(13/2))/(13*b^7) + (2*a^4*(a + b*x)^(15/2))/b^7 - (40*a^
3*(a + b*x)^(17/2))/(17*b^7) + (30*a^2*(a + b*x)^(19/2))/(19*b^7) - (4*a*(a + b*x)^(21/2))/(7*b^7) + (2*(a + b
*x)^(23/2))/(23*b^7)

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 x^6 (a+b x)^{9/2} \, dx &=\int \left (\frac {a^6 (a+b x)^{9/2}}{b^6}-\frac {6 a^5 (a+b x)^{11/2}}{b^6}+\frac {15 a^4 (a+b x)^{13/2}}{b^6}-\frac {20 a^3 (a+b x)^{15/2}}{b^6}+\frac {15 a^2 (a+b x)^{17/2}}{b^6}-\frac {6 a (a+b x)^{19/2}}{b^6}+\frac {(a+b x)^{21/2}}{b^6}\right ) \, dx\\ &=\frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 79, normalized size = 0.62 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (1024 a^6-5632 a^5 b x+18304 a^4 b^2 x^2-45760 a^3 b^3 x^3+97240 a^2 b^4 x^4-184756 a b^5 x^5+323323 b^6 x^6\right )}{7436429 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(1024*a^6 - 5632*a^5*b*x + 18304*a^4*b^2*x^2 - 45760*a^3*b^3*x^3 + 97240*a^2*b^4*x^4 - 184
756*a*b^5*x^5 + 323323*b^6*x^6))/(7436429*b^7)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 4.05, size = 143, normalized size = 1.13 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (1024 a^{11}-512 a^{10} b x+384 a^9 b^2 x^2-320 a^8 b^3 x^3+280 a^7 b^4 x^4-252 a^6 b^5 x^5+231 a^5 b^6 x^6+143 b^7 x^7 \left (3713 a^4+12770 a^3 b x+16830 a^2 b^2 x^2+10013 a b^3 x^3+2261 b^4 x^4\right )\right ) \sqrt {a+b x}}{7436429 b^7},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^7}{7}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^6*(a + b*x)^(9/2),x]')

[Out]

Piecewise[{{2 (1024 a ^ 11 - 512 a ^ 10 b x + 384 a ^ 9 b ^ 2 x ^ 2 - 320 a ^ 8 b ^ 3 x ^ 3 + 280 a ^ 7 b ^ 4
x ^ 4 - 252 a ^ 6 b ^ 5 x ^ 5 + 231 a ^ 5 b ^ 6 x ^ 6 + 143 b ^ 7 x ^ 7 (3713 a ^ 4 + 12770 a ^ 3 b x + 16830
a ^ 2 b ^ 2 x ^ 2 + 10013 a b ^ 3 x ^ 3 + 2261 b ^ 4 x ^ 4)) Sqrt[a + b x] / (7436429 b ^ 7), b != 0}}, a ^ (9
/ 2) x ^ 7 / 7]

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Maple [A]
time = 0.08, size = 85, normalized size = 0.67

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (323323 x^{6} b^{6}-184756 a \,x^{5} b^{5}+97240 a^{2} x^{4} b^{4}-45760 a^{3} b^{3} x^{3}+18304 a^{4} x^{2} b^{2}-5632 a^{5} x b +1024 a^{6}\right )}{7436429 b^{7}}\)	\(76\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {23}{2}}}{23}-\frac {4 a \left (b x +a \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {40 a^{3} \left (b x +a \right )^{\frac {17}{2}}}{17}+2 a^{4} \left (b x +a \right )^{\frac {15}{2}}-\frac {12 a^{5} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{6} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{7}}\)	\(85\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {23}{2}}}{23}-\frac {4 a \left (b x +a \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {40 a^{3} \left (b x +a \right )^{\frac {17}{2}}}{17}+2 a^{4} \left (b x +a \right )^{\frac {15}{2}}-\frac {12 a^{5} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{6} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{7}}\)	\(85\)
trager	\(\frac {2 \left (323323 b^{11} x^{11}+1431859 a \,b^{10} x^{10}+2406690 a^{2} b^{9} x^{9}+1826110 a^{3} b^{8} x^{8}+530959 a^{4} b^{7} x^{7}+231 a^{5} b^{6} x^{6}-252 a^{6} b^{5} x^{5}+280 a^{7} b^{4} x^{4}-320 a^{8} b^{3} x^{3}+384 b^{2} a^{9} x^{2}-512 a^{10} b x +1024 a^{11}\right ) \sqrt {b x +a}}{7436429 b^{7}}\)	\(131\)
risch	\(\frac {2 \left (323323 b^{11} x^{11}+1431859 a \,b^{10} x^{10}+2406690 a^{2} b^{9} x^{9}+1826110 a^{3} b^{8} x^{8}+530959 a^{4} b^{7} x^{7}+231 a^{5} b^{6} x^{6}-252 a^{6} b^{5} x^{5}+280 a^{7} b^{4} x^{4}-320 a^{8} b^{3} x^{3}+384 b^{2} a^{9} x^{2}-512 a^{10} b x +1024 a^{11}\right ) \sqrt {b x +a}}{7436429 b^{7}}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/b^7*(1/23*(b*x+a)^(23/2)-2/7*a*(b*x+a)^(21/2)+15/19*a^2*(b*x+a)^(19/2)-20/17*a^3*(b*x+a)^(17/2)+a^4*(b*x+a)^
(15/2)-6/13*a^5*(b*x+a)^(13/2)+1/11*a^6*(b*x+a)^(11/2))

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Maxima [A]
time = 0.27, size = 101, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {23}{2}}}{23 \, b^{7}} - \frac {4 \, {\left (b x + a\right )}^{\frac {21}{2}} a}{7 \, b^{7}} + \frac {30 \, {\left (b x + a\right )}^{\frac {19}{2}} a^{2}}{19 \, b^{7}} - \frac {40 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{3}}{17 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{4}}{b^{7}} - \frac {12 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{5}}{13 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{11 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/23*(b*x + a)^(23/2)/b^7 - 4/7*(b*x + a)^(21/2)*a/b^7 + 30/19*(b*x + a)^(19/2)*a^2/b^7 - 40/17*(b*x + a)^(17/
2)*a^3/b^7 + 2*(b*x + a)^(15/2)*a^4/b^7 - 12/13*(b*x + a)^(13/2)*a^5/b^7 + 2/11*(b*x + a)^(11/2)*a^6/b^7

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Fricas [A]
time = 0.31, size = 130, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (323323 \, b^{11} x^{11} + 1431859 \, a b^{10} x^{10} + 2406690 \, a^{2} b^{9} x^{9} + 1826110 \, a^{3} b^{8} x^{8} + 530959 \, a^{4} b^{7} x^{7} + 231 \, a^{5} b^{6} x^{6} - 252 \, a^{6} b^{5} x^{5} + 280 \, a^{7} b^{4} x^{4} - 320 \, a^{8} b^{3} x^{3} + 384 \, a^{9} b^{2} x^{2} - 512 \, a^{10} b x + 1024 \, a^{11}\right )} \sqrt {b x + a}}{7436429 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/7436429*(323323*b^11*x^11 + 1431859*a*b^10*x^10 + 2406690*a^2*b^9*x^9 + 1826110*a^3*b^8*x^8 + 530959*a^4*b^7
*x^7 + 231*a^5*b^6*x^6 - 252*a^6*b^5*x^5 + 280*a^7*b^4*x^4 - 320*a^8*b^3*x^3 + 384*a^9*b^2*x^2 - 512*a^10*b*x
+ 1024*a^11)*sqrt(b*x + a)/b^7

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Sympy [A]
time = 1.50, size = 257, normalized size = 2.02 \begin {gather*} \begin {cases} \frac {2048 a^{11} \sqrt {a + b x}}{7436429 b^{7}} - \frac {1024 a^{10} x \sqrt {a + b x}}{7436429 b^{6}} + \frac {768 a^{9} x^{2} \sqrt {a + b x}}{7436429 b^{5}} - \frac {640 a^{8} x^{3} \sqrt {a + b x}}{7436429 b^{4}} + \frac {80 a^{7} x^{4} \sqrt {a + b x}}{1062347 b^{3}} - \frac {72 a^{6} x^{5} \sqrt {a + b x}}{1062347 b^{2}} + \frac {6 a^{5} x^{6} \sqrt {a + b x}}{96577 b} + \frac {7426 a^{4} x^{7} \sqrt {a + b x}}{52003} + \frac {25540 a^{3} b x^{8} \sqrt {a + b x}}{52003} + \frac {1980 a^{2} b^{2} x^{9} \sqrt {a + b x}}{3059} + \frac {62 a b^{3} x^{10} \sqrt {a + b x}}{161} + \frac {2 b^{4} x^{11} \sqrt {a + b x}}{23} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{7}}{7} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**(9/2),x)

[Out]

Piecewise((2048*a**11*sqrt(a + b*x)/(7436429*b**7) - 1024*a**10*x*sqrt(a + b*x)/(7436429*b**6) + 768*a**9*x**2
*sqrt(a + b*x)/(7436429*b**5) - 640*a**8*x**3*sqrt(a + b*x)/(7436429*b**4) + 80*a**7*x**4*sqrt(a + b*x)/(10623
47*b**3) - 72*a**6*x**5*sqrt(a + b*x)/(1062347*b**2) + 6*a**5*x**6*sqrt(a + b*x)/(96577*b) + 7426*a**4*x**7*sq
rt(a + b*x)/52003 + 25540*a**3*b*x**8*sqrt(a + b*x)/52003 + 1980*a**2*b**2*x**9*sqrt(a + b*x)/3059 + 62*a*b**3
*x**10*sqrt(a + b*x)/161 + 2*b**4*x**11*sqrt(a + b*x)/23, Ne(b, 0)), (a**(9/2)*x**7/7, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (101) = 202\).
time = 0.01, size = 1278, normalized size = 10.06 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{23} \sqrt {a+b x} \left (a+b x\right )^{11}-\frac {11}{21} \sqrt {a+b x} \left (a+b x\right )^{10} a+\frac {55}{19} \sqrt {a+b x} \left (a+b x\right )^{9} a^{2}-\frac {165}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a^{3}+22 \sqrt {a+b x} \left (a+b x\right )^{7} a^{4}-\frac {462}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{5}+42 \sqrt {a+b x} \left (a+b x\right )^{5} a^{6}-\frac {110}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{7}+\frac {165}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{8}-11 \sqrt {a+b x} \left (a+b x\right )^{2} a^{9}+\frac {11}{3} \sqrt {a+b x} \left (a+b x\right ) a^{10}-\sqrt {a+b x} a^{11}\right )}{b^{11}}+\frac {10 a b^{4} \left (\frac {1}{21} \sqrt {a+b x} \left (a+b x\right )^{10}-\frac {10}{19} \sqrt {a+b x} \left (a+b x\right )^{9} a+\frac {45}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a^{2}-8 \sqrt {a+b x} \left (a+b x\right )^{7} a^{3}+\frac {210}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{4}-\frac {252}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{5}+\frac {70}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{6}-\frac {120}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{7}+9 \sqrt {a+b x} \left (a+b x\right )^{2} a^{8}-\frac {10}{3} \sqrt {a+b x} \left (a+b x\right ) a^{9}+\sqrt {a+b x} a^{10}\right )}{b^{10}}+\frac {20 a^{2} b^{3} \left (\frac {1}{19} \sqrt {a+b x} \left (a+b x\right )^{9}-\frac {9}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a+\frac {12}{5} \sqrt {a+b x} \left (a+b x\right )^{7} a^{2}-\frac {84}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{3}+\frac {126}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{4}-14 \sqrt {a+b x} \left (a+b x\right )^{4} a^{5}+12 \sqrt {a+b x} \left (a+b x\right )^{3} a^{6}-\frac {36}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{7}+3 \sqrt {a+b x} \left (a+b x\right ) a^{8}-\sqrt {a+b x} a^{9}\right )}{b^{9}}+\frac {20 a^{3} b^{2} \left (\frac {1}{17} \sqrt {a+b x} \left (a+b x\right )^{8}-\frac {8}{15} \sqrt {a+b x} \left (a+b x\right )^{7} a+\frac {28}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{2}-\frac {56}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{3}+\frac {70}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{4}-8 \sqrt {a+b x} \left (a+b x\right )^{3} a^{5}+\frac {28}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{6}-\frac {8}{3} \sqrt {a+b x} \left (a+b x\right ) a^{7}+\sqrt {a+b x} a^{8}\right )}{b^{8}}+\frac {10 a^{4} b \left (\frac {1}{15} \sqrt {a+b x} \left (a+b x\right )^{7}-\frac {7}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a+\frac {21}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{2}-\frac {35}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{3}+5 \sqrt {a+b x} \left (a+b x\right )^{3} a^{4}-\frac {21}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{5}+\frac {7}{3} \sqrt {a+b x} \left (a+b x\right ) a^{6}-\sqrt {a+b x} a^{7}\right )}{b^{7}}+\frac {2 a^{5} \left (\frac {1}{13} \sqrt {a+b x} \left (a+b x\right )^{6}-\frac {6}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{2}-\frac {20}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{3}+3 \sqrt {a+b x} \left (a+b x\right )^{2} a^{4}-2 \sqrt {a+b x} \left (a+b x\right ) a^{5}+\sqrt {a+b x} a^{6}\right )}{b^{6}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x)

[Out]

2/66927861*(22287*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^
(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*a^5/b^6 + 52003*(429
*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(
b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*a^4/b^6 +
6118*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2
)*a^3 + 850850*(b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 291720*(b*x + a
)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*a^3/b^6 + 2898*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 5
54268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 3233230*(b*x + a)^(
9/2)*a^5 + 2771340*(b*x + a)^(7/2)*a^6 - 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8 - 230945*sqr
t(b*x + a)*a^9)*a^2/b^6 + 345*(46189*(b*x + a)^(21/2) - 510510*(b*x + a)^(19/2)*a + 2567565*(b*x + a)^(17/2)*a
^2 - 7759752*(b*x + a)^(15/2)*a^3 + 15668730*(b*x + a)^(13/2)*a^4 - 22221108*(b*x + a)^(11/2)*a^5 + 22632610*(
b*x + a)^(9/2)*a^6 - 16628040*(b*x + a)^(7/2)*a^7 + 8729721*(b*x + a)^(5/2)*a^8 - 3233230*(b*x + a)^(3/2)*a^9
+ 969969*sqrt(b*x + a)*a^10)*a/b^6 + 33*(88179*(b*x + a)^(23/2) - 1062347*(b*x + a)^(21/2)*a + 5870865*(b*x +
a)^(19/2)*a^2 - 19684665*(b*x + a)^(17/2)*a^3 + 44618574*(b*x + a)^(15/2)*a^4 - 72076158*(b*x + a)^(13/2)*a^5
+ 85180914*(b*x + a)^(11/2)*a^6 - 74364290*(b*x + a)^(9/2)*a^7 + 47805615*(b*x + a)^(7/2)*a^8 - 22309287*(b*x
+ a)^(5/2)*a^9 + 7436429*(b*x + a)^(3/2)*a^10 - 2028117*sqrt(b*x + a)*a^11)/b^6)/b

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Mupad [B]
time = 0.03, size = 101, normalized size = 0.80 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{23/2}}{23\,b^7}+\frac {2\,a^6\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}-\frac {12\,a^5\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{15/2}}{b^7}-\frac {40\,a^3\,{\left (a+b\,x\right )}^{17/2}}{17\,b^7}+\frac {30\,a^2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^7}-\frac {4\,a\,{\left (a+b\,x\right )}^{21/2}}{7\,b^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(23/2))/(23*b^7) + (2*a^6*(a + b*x)^(11/2))/(11*b^7) - (12*a^5*(a + b*x)^(13/2))/(13*b^7) + (2*a^
4*(a + b*x)^(15/2))/b^7 - (40*a^3*(a + b*x)^(17/2))/(17*b^7) + (30*a^2*(a + b*x)^(19/2))/(19*b^7) - (4*a*(a +
b*x)^(21/2))/(7*b^7)

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