Optimal. Leaf size=196 \[ -\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 b^{7/4} d \sqrt {a+b x}}+\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b} \]
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Rubi [A] time = 0.22, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {50, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}+\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/4}}{\sqrt {a+b x}} \, dx &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}+\frac {(3 (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{5 b}\\ &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}+\frac {(12 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b d}\\ &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}-\frac {\left (12 (b c-a d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b^{3/2} d}+\frac {\left (12 (b c-a d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b^{3/2} d}\\ &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}-\frac {\left (12 (b c-a d)^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b^{3/2} d \sqrt {a+b x}}+\frac {\left (12 (b c-a d)^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b^{3/2} d \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}-\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}+\frac {\left (12 (b c-a d)^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 b^{3/2} d \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}+\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}-\frac {12 (b c-a d)^{7/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 0.36 \[ \frac {2 \sqrt {a+b x} (c+d x)^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{\frac {3}{4}}}{\sqrt {b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {3}{4}}}{\sqrt {b x +a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^{3/4}}{\sqrt {a+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\sqrt {a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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